About this Journal Submit a Manuscript Table of Contents
ISRN Discrete Mathematics
VolumeΒ 2011Β (2011), Article IDΒ 108509, 21 pages
http://dx.doi.org/10.5402/2011/108509
Research Article

On the Spectrum of Threshold Graphs

Mathematics Department, Faculty of Science, University of Malta, Msida MSD 2080, Malta

Received 29 September 2011; Accepted 3 November 2011

Academic Editors: F. Ergun, P. Feinsilver, Y. Hou, S. Y. Song, and N. I. Trinajstić

Copyright Β© 2011 Irene Sciriha and Stephanie Farrugia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The antiregular connected graph on π‘Ÿ vertices is defined as the connected graph whose vertex degrees take the values of π‘Ÿβˆ’1 distinct positive integers. We explore the spectrum of its adjacency matrix and show common properties with those of connected threshold graphs, having an equitable partition with a minimal number π‘Ÿ of parts. Structural and combinatorial properties can be deduced for related classes of graphs and in particular for the minimal configurations in the class of singular graphs.

1. Introduction

A graph 𝐺=𝐺(𝒱,β„°) of order 𝑛 has a labelled vertex set 𝒱={1,2,…,𝑛} containing 𝑛 vertices and a set β„° of π‘š edges consisting of unordered pairs of the vertices. When a subset 𝒱1 of 𝒱 is deleted, the edges incident to 𝒱1 are also deleted. The subgraph πΊβˆ’π’±1 of 𝐺 is said to be an induced subgraph of 𝐺. The subgraph of 𝐺 obtained by deleting a particular vertex 𝑣 is simply denoted by πΊβˆ’π‘£. The cycle and the complete graph on 𝑛 vertices are denoted by 𝐢𝑛 and 𝐾𝑛, respectively.

The graphs we consider are simple, that is, without loops or multiple edges. We use bold face, say 𝐆, to denote the 0-1-adjacency matrix of the graph bearing the same name 𝐺, where the 𝑖𝑗th entry of the symmetric matrix 𝐆 is 1 if {𝑖,𝑗}βˆˆβ„° and 0 otherwise. We note that the graph 𝐺 is determined, up to isomorphism, by 𝐆. The adjacency matrix 𝐆𝐢 of the complement 𝐺𝐢 of 𝐺 is π‰βˆ’πˆβˆ’π†, where each entry of 𝐉 is one and 𝐈 is the identity matrix. The degree of a vertex 𝑖 is the number of nonzero entries in the 𝑖th row of 𝐆.

The disconnected graph with two components 𝐺1 and 𝐺2 is their disjoint union, denoted by 𝐺1Μ‡βˆͺ𝐺2. For π‘Ÿβ‰₯2, the graph π‘ŸπΊ is the disconnected graph with π‘Ÿ components, where each component is isomorphic to 𝐺. The join 𝐺1βˆ‡πΊ2 of 𝐺1 and 𝐺2 is (𝐺𝐢1Μ‡βˆͺ𝐺𝐢2)𝐢.

For the linear transformation 𝐆, the 𝑛 real numbers {πœ†}satisfying 𝐆𝐱=πœ†π± for some nonzero vector π±βˆˆβ„π‘› are said to be eigenvalues of 𝐆 and form the spectrum of 𝐺. They are the solutions of the characteristic polynomial πœ™(𝐺,πœ†) of 𝐆, defined as the polynomial det(πœ†πˆβˆ’π†) in πœ†. The subspace ker𝐆 of 𝐑𝑛 that maps to zero under 𝐆 is said to be the nullspace of 𝐆. A graph 𝐺 is said to be singular of nullity πœ‚ if the dimension of ker(𝐆) is πœ‚. The nonzero vectors, π‘₯βˆˆβ„π‘›, in the nullspace, termed kernel eigenvectors of 𝐺, satisfy 𝐆𝐱=𝟎. We note that the multiplicity of the eigenvalue zero is πœ‚. If there exists a kernel eigenvector of 𝐆 with no zero entries, then 𝐺 is said to be a core graph. The cycle 𝐢4 on four vertices is a core graph of nullity two with a kernel eigenvector (1,1,βˆ’1,βˆ’1)𝑑 for the usual labelling of the vertices round the cycle. A core graph of nullity one is said to be a nut graph [1]. A minimal configuration for a particular core, to be defined formally in Section 6, is intuitively a graph of nullity one with a minimal number of vertices and edges for that core.

The distinct eigenvalues πœ‡1,πœ‡2,…,πœ‡π‘,1≀𝑝≀𝑛, which have an associated eigenvector not orthogonal to 𝐣 (the vector with each entry equal to one) are said to be main. We denote the remaining distinct eigenvalues by πœ‡π‘+1,…,πœ‡π‘ ,𝑠≀𝑛, and refer to them as nonmain. By the Perron-Frobenius theorem [2, page 6] the maximum eigenvalue of the adjacency matrix of a connected graph has an associated eigenvector (termed the Perron vector) with all its entries positive. Therefore, at least one eigenvalue of a graph is main.

A cograph, or complement-reducible graph, is a graph that can be generated from the single-vertex graph 𝐾1 by complementation and disjoint union. Threshold graphs are a subclass of cographs. They were first introduced in 1977 by ChvΓ‘tal and Hammer in connection with the equivalence between set packing and knapsack problems [3] and independently, in the same year, by Henderson and Zalcstein for parallel systems in computer programming [4]. It is surprising that they kept being rediscovered in different contexts leading to several equivalent definitions. The most useful for our purposes are two, given below: one in terms of their forbidden induced subgraphs and the other in terms of their degree sequence [5, 6]. For the latter definition, the graph partition Ξ  of 2π‘š into parts equal to the vertex degrees {𝜌1,𝜌2,…,πœŒπ‘›} is needed. The array of boxes 𝐹(Ξ ), known as a Ferrers/Young diagram for the monotonic nonincreasing sequence Ξ ={𝜌1,𝜌2,…,πœŒπ‘›} consists of 𝑛 rows of πœŒπ‘– boxes as 𝑖 runs successively from 1 to 𝑛. Threshold graphs are characterized by a particular shape of the Ferrers/Young diagram (see Figure 4), which will be described in Section 3.4.

Definition 1.1. (i) A threshold graph is a graph with no induced subgraphs isomorphic to any of the following subgraphs on four vertices: the path 𝑃4, the cycle 𝐢4 and the two copies 2𝐾2 of the complete graph 𝐾2 on two vertices. It is said to be 𝑃4-, 𝐢4-, and 2𝐾2-free.
Equivalently, (ii) if the monotonic nonincreasing degree sequence, Ξ ={𝜌1,𝜌2,…,πœŒπ‘›}, of a graph 𝐺 is represented by the rows of a Ferrers/Young diagram 𝐹(Ξ ), where the length of the principal square of 𝐹(Ξ ) is 𝑓(Ξ ) and the lengths {πœ‹βˆ—π‘˜βˆΆ1β‰€π‘˜β‰€π‘“(Ξ )} of the columns of 𝐹(Ξ ) satisfy πœ‹βˆ—π‘˜=πœŒπ‘˜+1, then 𝐺 is said to be a threshold graph [7, Lemma 7.23].

If the parts of a threshold graph partition of 2π‘š are all equal, then the graph is regular and corresponds to the complete graph. If, on the other hand, there are as many distinct sizes of the parts of a threshold graph partition of 2π‘š as possible, then the graph is said to be antiregular. Recall that at least two vertices in a graph have the same degree.

Definition 1.2. An antiregular graph on π‘Ÿ vertices is defined as a threshold graph whose vertex degrees take as many different values as possible, that is, π‘Ÿβˆ’1 distinct nonnegative integral values.

Definition 1.3. The partition 𝒱1Μ‡βˆͺ𝒱2Μ‡Μ‡βˆͺβ‹―βˆͺπ’±π‘Ÿ of the vertex set 𝒱 of a graph 𝐺 is said to be an equitable partition if, for all 𝑖,π‘—βˆˆ{1,2,…,π‘Ÿ}, the number of neighbours in 𝒱𝑗 of a vertex in 𝒱𝑖 depends only on the choice of 𝑖 and 𝑗.

The overall aim of this paper is to explore the spectrum of its adjacency matrix and show common properties with those of connected threshold graphs, having an equitable partition with a minimal number π‘Ÿ of parts.

The paper is organised as follows. In Section 2, cographs are reviewed and made use of in Section 3 to determine a particular representation of a threshold graph that has earned it the name of nested split graph. We also present various other representations that are used selectively to simplify our proofs. In Section 4, a procedure that transforms the Ferrers/Young diagram into the adjacency matrix of the threshold graph for a particular vertex labelling is given. The structures of the graph and of its underlying antiregular graph are also compared.

Our main results are as follows.(i)In Section 5, the Ferrers/Young diagram comes in use to explore the nullspace of a threshold graph.(ii)In Section 6, we show that all minimal configurations on at least five vertices have the subgraph 𝑃4 induced.(iii)We show in Section 7 that the spectrum of a connected threshold graph 𝐺 and its underlying antiregular graph show common characteristics. All the eigenvalues other than 0 and βˆ’1 are main and each main eigenvalue contributes to the number of walks. Moreover, the spectrum of its quotient graph 𝐺/Ξ  consists precisely of the main eigenvalues of 𝐺. The characteristic polynomial of 𝐺/Ξ  is reducible over the integers (i.e., it has polynomial factors) for certain threshold graphs 𝐺.(iv)We end with a discussion, in Section 8, on the variation in the sign pattern of the spectrum as vertices are added to a threshold graph to produce another threshold graph.

2. Cographs

A cograph is the union or the join of subgraphs of the form (β‹―((π‘Ÿ1𝐾1)𝐢̇βˆͺ(π‘Ÿ2𝐾1))𝐢̇βˆͺβ‹―Μ‡βˆͺ(π‘Ÿπ‘ πΎ1)𝐢), where π‘Ÿπ‘–βˆˆβ„€+βˆͺ{0}, for all 𝑖. Therefore, the family of cographs is the smallest class of graphs that includes 𝐾1 and is closed under complementation and disjoint union. It is well known that no cograph on at least four vertices has 𝑃4 as an induced subgraph [8]. In fact cographs can also be characterized as 𝑃4-free graphs.

Cographs have received much attention since the 1970s. They were discovered independently by many authors including Jung [9] in 1978, Lerchs [10] in 1971 and, Seinsche [11] and Sumner [12], both in 1974. For a more detailed treatment of cographs, see [8].

Connected graphs, which are 2𝐾2-, 𝑃4-, and 𝐢4-free, necessarily have a dominating vertex, that is, a vertex adjacent to all the other vertices of the graph. Thus, all connected threshold graphs have a dominating vertex.

By construction, a connected cograph also has a dominating vertex. Therefore, its complement has at least one isolated vertex. A necessary condition for a connected graph to have a connected complement is that it has 𝑃4 as an induced subgraph [7, Theorem 1.19]. The set of cographs and the class of graphs with a connected complement are disjoint as sets. However, if the graph 𝐻 is 𝑃4Μ‡βˆͺ𝐾1, then both 𝐻 and 𝐻𝐢 have 𝑃4-induced. Thus there exist connected graphs that are neither 𝑃4 free nor have a connected complement.

Recall that 𝐺1βˆ‡πΊ2=(𝐺𝐢1Μ‡βˆͺ𝐺𝐢2)𝐢. Hence, cographs are also characterized as the smallest class of graphs that includes 𝐾1 and is closed under join and disjoint union. On this definition of cographs, the proofs in [13], of the result that cographs are polynomial reconstructible from the deck of characteristic polynomials of the one-vertex deleted subgraphs, are based.

A cograph can be represented uniquely by a cotree, as explained in [14] and later in [13]. Figure 1 shows the cotree 𝑇𝐺 of the cograph 𝐺. The vertices βŠ•, βŠ—, and β€’ of a cotree represent the disjoint union, the join, and the vertices of the cograph, respectively. For simplicity we say that the terminal vertices of 𝑇𝐺 are vertices of 𝐺. The cotree 𝑇𝐺 is a rooted tree and only the terminal vertices represent the cograph vertices. An interior vertex βŠ• or βŠ— of 𝑇𝐺 represents the subgraph of 𝐺 induced by its terminal successors. The immediate successors of βŠ• can be cograph vertices β€’ or βŠ—. Similarly the immediate successors of βŠ— can be cograph vertices β€’ or βŠ•. Therefore, the interior vertices of 𝑇𝐺 on a (oriented) path descending from the root to a terminal vertex of 𝑇𝐺 are a sequence of alternating βŠ— and βŠ•.

fig1
Figure 1: Cotree 𝑇𝐺 for the cograph 𝐺.

3. Representations of Threshold Graphs

In this section we present some of the various representations of threshold graphs. Collectively, they provide a wealth of information that determine combinatorial properties of these graphs. We start with the cotree representation as in the previous section. There are certain restrictions on the structure of a cotree in the case when a cograph is a threshold graph.

We give a proof to the following result quoted in [13].

Lemma 3.1. If a cograph 𝐺 is also a threshold graph, then each interior vertex of 𝑇𝐺 has at most one interior vertex as an immediate successor.

Proof. A threshold graph 𝐺 is 𝑃4-free and therefore is a cograph which can be represented by a cotree 𝑇𝐺. Note that 𝑃4 cannot be represented as a cotree. In a threshold graph, there are no induced subgraphs isomorphic to 𝐢4 or to 2𝐾2. Therefore, the configurations in Figure 2(a) representing 𝐢4 and 2(b) representing 2𝐾2 as cotrees are not allowed in the cotree 𝑇𝐺 corresponding to a threshold graph 𝐺. We deduce that the number of interior vertices which are immediate successors of an interior vertex is less than two, as required.

fig2
Figure 2: Representations of 𝐢4 and 2𝐾2 in a cotree.

We now present various other representations of threshold graphs that are used in the proofs that follow.

3.1. Cotrees of Nested Split Graphs

A caterpillar is a tree in which the removal of all terminal vertices (i.e., those of degree 1) gives a path. The following result follows immediately from Lemma 3.1.

Corollary 3.2. The cotree of a threshold graph is a caterpillar.

The vertex set of a split graph is partitioned into two subsets, one of which is a clique (inducing a complete subgraph) and the other a coclique or an independent set (inducing the empty graph with no edges). Because of its structure, a threshold graph is also referred to as a nested split graph.

The first vertex labelling (which we will refer to as Lab1) of a threshold graph is according to its construction. Starting from 𝐾1 (vertex 1a), the graph in Figure 3 is (((((((((𝐾1Μ‡βˆͺ𝐾1)βˆ‡πΎ1)βˆ‡πΎ1)Μ‡βˆͺ𝐾1)βˆ‡πΎ1)βˆ‡πΎ1)Μ‡βˆͺ𝐾1)βˆ‡πΎ1)βˆ‡πΎ1) coded as ((((((𝐾1Μ‡βˆͺ𝐾1)βˆ‡2𝐾1)Μ‡βˆͺ𝐾1)βˆ‡2𝐾1)Μ‡βˆͺ𝐾1)βˆ‡2𝐾1) to avoid repetitions of successive joins or unions. Therefore, according to the vertex labelling in Figure 3, 𝐺 is Μ‡Μ‡Μ‡(((((((((1π‘Žβˆͺ1𝑏)βˆ‡2π‘Ž)βˆ‡2𝑏)βˆͺ3π‘Ž)βˆ‡4π‘Ž)βˆ‡4𝑏)βˆͺ5π‘Ž)βˆ‡6π‘Ž)βˆ‡6𝑏). The cotree 𝑇𝐺 represents the threshold graph 𝐺 drawn next to it in a way so as to emphasise the nested split graph structure of 𝐺, where the circumscribed vertices labelled 1 represent the subgraph induced by the vertices 1a and 1b, and similarly for the other circumscribed subsets of vertices.

fig3
Figure 3: Cotree 𝑇𝐺 and the nested structure of the threshold graph 𝐺.
fig4
Figure 4: The Ferrers/Young diagram for a threshold graph.

In 𝑇𝐺, the terminal vertices {β€’} which are immediate successors of a vertex βŠ— form a clique (inducing a complete subgraph) whereas those immediately succeeding a vertex βŠ• form a coclique (inducing a subgraph without edges). A line in 𝐺 joining 𝑅 and 𝑆, which are circumscribed cliques or cocliques, means that each vertex of 𝑅 is adjacent to each vertex of 𝑆.

3.2. Minimal Equitable Partition of the Vertex Set

Our labelling of the π‘Ÿ parts in the equitable partition of the vertices of a connected threshold graph 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) follows the addition of the vertices in the construction in order, namely, Μ‡βˆͺ((β‹―(π‘Ž1𝐾1βˆ‡π‘Ž2𝐾1)Μ‡βˆͺπ‘Ž3𝐾1)βˆ‡β‹―π‘Žπ‘ŸπΎ1) according to the coded representation of the graph in Figure 3. Then, the nested structure of the threshold graph becomes clear. The parts are cliques or cocliques of size π‘Žπ‘– for 1β‰€π‘–β‰€π‘Ÿ. For a minimal value of π‘Ÿ, Ξ  is said to be a nondegenerate equitable partition for the nondegenerate representation 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ). All other equitable partitions of the vertex set are refinements of Ξ  with a larger number of parts, when an equitable partition and the corresponding representation 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) are said to be degenerate. Unless otherwise stated we will assume that equitable vertex partitions and representations are nondegenerate. In particular π‘Ž1β‰ 1.

According to our labelling convention (Lab1) for 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) as in Figure 3, a threshold graph 𝐺 whose cotree 𝑇𝐺 has root βŠ— is connected. If π‘Ÿ is even, then π‘Ž1 is associated with a coclique, whereas, for π‘Ÿ odd, π‘Ž1 is associated with a clique. It follows that the monotonic nonincreasing vertex degree sequence of 𝐺 will be associated with π‘Žπ‘Ÿ,π‘Žπ‘Ÿβˆ’2,…,π‘Ž2,π‘Ž1,π‘Ž3,…,π‘Žπ‘Ÿβˆ’1 in that order if π‘Ÿ is even and π‘Žπ‘Ÿ,π‘Žπ‘Ÿβˆ’2,…,π‘Ž1,π‘Ž2,π‘Ž4,…,π‘Žπ‘Ÿβˆ’1 in that order if π‘Ÿ is odd. By convention therefore, for a nondegenerate equitable partition, π‘Žπ‘–β‰₯1 for 2β‰€π‘–β‰€π‘Ÿβˆ’1 and π‘Ž1β‰₯2. According to this representation, the graph of Figure 3 has the nondegenerate representation 𝐢(2,2,1,2,1,2).

3.3. The Binary Code of a Threshold Graph

For the purposes of inputting an 𝑛-vertex-threshold graph to be processed in a computer program, the graph is encoded as a string of π‘›βˆ’1 bits. The graph is represented as a sequence of 0 and 1 entries where 0 represents the addition of an isolated vertex and 1 represents the addition of a dominating vertex in the construction of the graph, staring from 𝐾1, as described above.

The graph of Figure 3 is encoded as (011011011).

3.4. Degree Sequence

The last representation of a threshold graph that we now give is constructed from the degree sequence. Following Definition 1.1(ii), let 𝐹(Ξ ) be the Ferrers/Young diagram (Figure 4) for the nonincreasing degree sequence giving a vertex partition Ξ ={𝜌1,𝜌2,…,πœŒπ‘›} of 2π‘š for an 𝑛-vertex graph. The largest principal square of boxes in 𝐹(Ξ ) is termed the Durfee square and 𝑓(Ξ ) denotes the size of the Durfee square (i.e., the length of a side of the Durfee square). A graph is graphical if and only if πœ‹βˆ—π‘–β‰₯πœŒπ‘–+1 for 1≀𝑖≀𝑓(Ξ ) [15].

It is well known that there exist nonisomorphic graphs with the same degree sequence. A graph determined, up to isomorphism, by its degree sequence is said to be a unigraph.

Lemma 3.3 (see [7, Theorem 7.30]). A threshold graph is a unigraph.

The degree sequence Π of a threshold graph also produces a particular structure of the Ferrers/Young diagram 𝐹(Π), shown in Figure 4.

Lemma 3.4 (see [7]). For a threshold graph, 𝐹(Ξ ) consists of four blocks 𝑃, 𝑄, 𝑅, and its transpose 𝑅𝑑, where 𝑃 is the Durfee square, 𝑄 is the (𝑓(Ξ )+1)th row of 𝐹(Ξ ) of length 𝑓(Ξ ), and 𝑅 is the array of boxes left after removing the Durfee square from the first 𝑓(Ξ ) rows of 𝐹(Ξ ).

4. The Structure of Threshold Graphs

An interesting algorithm was presented in [15] to construct a threshold graph. The adjacency list π‘Žπ‘‘π‘—πΏπ‘–π‘ π‘‘ of the graph, that is the list of neighbours of each vertex, is in fact obtained by filling in the boxes of the 𝑖th row in 𝐹(Ξ ) with consecutive integers starting from 1, but skipping 𝑖. By Lemma 3.3, 𝐹(Ξ ) gives a unique threshold graph, up to isomorphism and therefore provides a canonical vertex labelling. We now present a procedure to produce the adjacency matrix of the labelled threshold graph corresponding to adjList from 𝐹(Ξ ). We note that this gives us the second labelling, Lab2, in order of the nonincreasing degree sequence and therefore different from Lab1 used for 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ).

Theorem 4.1. The 𝑛×𝑛 adjacency matrix 𝐆 of a threshold graph 𝐺 is obtained from its Ferrers/Young diagram 𝐹(Ξ ), representing the degree sequence of a 𝑛-vertex graph, as follows. The 𝑖th box is inserted in each 𝑖th row and filled with a zero entry. The rest of the existing boxes are filled with the entry 1. Boxes are now inserted so that a 𝑛×𝑛 array of boxes is obtained. Each of the remaining empty boxes is filled with zero. The 𝑛×𝑛 array of 0-1-numbers obtained is the adjacency matrix 𝐆.

The rows and columns of the adjacency matrix constructed in Theorem 4.1 are indexed according to the nonincreasing degree sequence. If, for a threshold graph, each of the boxes of the 𝑖th row in 𝐹(Ξ ) is filled with 𝑖 to obtain 𝐻(Ξ ), then the adjacency list adjList of the graph is just a rearrangement of the entries of 𝐻(Ξ ) since, by Definition 1.1, πœ‹βˆ—π‘˜=πœŒπ‘˜+1. Due to the shape of the nonzero part, the adjacency matrix is said to have β€œa stepwise” form [16, 17].

4.1. The Antiregular Graph

The antiregular graph π΄π‘Ÿ may be considered to be the smallest threshold graph for an equitable vertex partition having a given number (π‘Ÿβˆ’1) of parts.

Definition 4.2. An antiregular graph π΄π‘Ÿ on π‘Ÿ vertices is a graph whose vertex degrees take the values of π‘Ÿβˆ’1 distinct (nonnegative) integers.

We shall use the π‘Ÿ-vertex connected antiregular graph π΄π‘Ÿ with the largest number (π‘Ÿβˆ’1) of parts in its equitable partition, having degenerate representation 𝐢(1,1,…,1) using Lab1. Any part can be expanded to produce a threshold graph 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ), taking care to preserve the nested split structure. The connected antiregular graph π΄π‘Ÿ with degenerate equitable partition into π‘Ÿ parts is adopted as the underlying graph of a connected threshold graph for an equitable vertex partition with π‘Ÿ parts.

Lemma 4.3. An induced subgraph 𝐻 of 𝐺=𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) is 𝐢(𝑏1,𝑏2,…,π‘π‘Ÿ), where 0β‰€π‘π‘–β‰€π‘Žπ‘– for 1≀𝑖≀𝑠.

Proof. When π‘Žπ‘–β‰ π‘π‘– for at least one value of 𝑖, to produce 𝐻, vertices are deleted from the part of size π‘Žπ‘– in the equitable partition of 𝒱𝐺. This procedure produces an induced subgraph at each stage and it is repeated until 𝑏𝑖 is reached for each 𝑖.

The threshold graph 𝐢(1,1,…,1) having π‘Ÿ parts, where each part is of size 1, is the degenerate form of π΄π‘Ÿ. Its nondegenerate form, consistent with the cotree representations of threshold graphs, is 𝐢(2,1,…,1,1) having π‘Ÿβˆ’1 parts, with only the first part of size 2. As an immediate consequence of Lemma 4.3 we have the following.

Corollary 4.4. The connected antiregular graph 𝐢(2,1,1,…,1), having π‘Ÿβˆ’1 parts, with degenerate representation 𝐢(1,1,1,…,1), having π‘Ÿ parts, is an induced subgraph of 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) where 1β‰€π‘Žπ‘– for 2β‰€π‘–β‰€π‘Ÿ and π‘Ž1β‰₯2.

On taking the complement of 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) or on deleting a dominating vertex when π‘Žπ‘Ÿ=1, a disconnected graph is obtained (see Figures 5 and 6).

fig5
Figure 5: The threshold graph 𝐺=𝐢(π‘Ž1,π‘Ž2,…,π‘Ž7), 𝐺𝐢 and πΊβˆ’π‘£ for π‘Ž7=1.
fig6
Figure 6: The threshold graph 𝐺=𝐢(π‘Ž1,π‘Ž2,…,π‘Ž8), 𝐺𝐢 and πΊβˆ’π‘£ for π‘Ž8=1 (Lab1).

Proposition 4.5. Let 𝑣 be the dominating vertex of π΄π‘Ÿ. Then, (i)π΄π‘Ÿβˆ’π‘£ is 𝐾1Μ‡βˆͺπ΄π‘Ÿβˆ’2 and (ii)π΄πΆπ‘Ÿ=𝐾1Μ‡βˆͺπ΄π‘Ÿβˆ’1.

Figures 5 and 6, respectively, show the threshold graphs with underlying 𝐴7 and 𝐴8, their complements, and the 𝑣-deleted subgraphs when 𝑣 is the only dominating vertex. The corresponding representations of 𝐴7 and 𝐴8 are 𝐢(2,1,1,1,1,1) and 𝐢(2,1,1,1,1,1,1), respectively.

Proposition 4.6. The binary codes for the connected antiregular graphs 𝐴2π‘˜ and 𝐴2π‘˜+1 are, respectively, the (2π‘˜βˆ’1)-string (1010β‹―1) and the 2π‘˜-string (010β‹―1) with alternating 0 and 1 entries.

Since the binary code follows the construction of π΄π‘Ÿ algorithmically, we have the following.

Corollary 4.7. The construction of connected antiregular graphs is as follows: for π‘˜βˆˆβ„€+:𝐴2π‘˜=(β‹―(𝐾1βˆ‡πΎ1)Μ‡βˆͺ𝐾1)βˆ‡β‹―)βˆ‡πΎ1, 𝐴2π‘˜+1=(β‹―(𝐾1Μ‡βˆͺ𝐾1)βˆ‡πΎ1)Μ‡βˆͺβ‹―)βˆ‡πΎ1.

4.2. The Complement of a Threshold Graph

The complement of a connected threshold graph 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) is disconnected and is denoted by 𝐷(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) (see Figure 3). The following result is deduced from the construction of the complement.

Proposition 4.8 (see [18]). The cotree 𝑇𝐺𝐢 of the complement 𝐺𝐢=𝐷(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) of 𝐺=𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) is obtained from 𝑇𝐺 by changing the interior vertices from βŠ— to βŠ• and viceversa.

Corollary 4.9. The complement 𝐺𝐢 of the connected threshold graph 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ), is the disconnected threshold graph 𝐷(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) isomorphic to 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿβˆ’1)Μ‡βˆͺπ‘Žπ‘ŸπΎ1.

Proof. Since 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) is connected, its cotree has βŠ— as a root. Therefore, by Proposition 4.8, the cotree 𝐷(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) has βŠ• as a root, and therefore it has coclique πΎπ‘Žπ‘Ÿ.

Proposition 4.10. The binary string coding of the threshold graph 𝐢(π‘Ž1,π‘Ž2,…,π‘Ž2π‘˜), with the underlying graph 𝐴2π‘˜, is the 2π‘˜-string (0π‘Ž1βˆ’11π‘Ž2β‹―1π‘Ž2π‘˜) of 0 and 1 entries. (The superscripts denote repetition; 1π‘Žπ‘– denotes the substring 111… with 1 repeated π‘Žπ‘– times).

Similarly the binary string coding of the threshold graph 𝐢(π‘Ž1,π‘Ž2,…,π‘Ž2π‘˜+1), with underlying graph 𝐴2π‘˜+1, is the 2π‘˜+1-string (1π‘Ž1βˆ’10π‘Ž2β‹―1π‘Ž2π‘˜+1).

5. The Nullity of Threshold Graphs

A pair of duplicate vertices of a graph are nonadjacent and have common neighbours, whereas a pair of coduplicate vertices are adjacent and have common neighbours. The rows of the adjacency matrix corresponding to duplicate vertices are identical and for those of coduplicate vertices π‘˜ and β„Ž, the π‘˜th and β„Žth rows differ only in the π‘˜th and β„Žth entries. It follows that both duplicates and coduplicates produce the eigenvector with only two nonzero entries, namely, 1 and βˆ’1, at positions corresponding to the pair of vertices, with corresponding eigenvalue 0 and βˆ’1, respectively.

Remark 5.1. In this section we adopt the vertex labelling Lab2 of a threshold graph induced by the Ferrers/Young diagram in accordance with the procedure to form the β€œstepwise” adjacency matrix presented in Theorem 4.1.

A graph with duplicates is often considered as having repeated vertices and therefore redundant properties. We call the induced subgraph of a graph obtained by removing repeated vertices canonical.

Theorem 5.2. An upper bound for the nullity πœ‚(𝐺) of the adjacency matrix of a threshold graph is π‘›βˆ’π‘“(Ξ )βˆ’1.

Proof. When the adjacency matrix 𝐆 is obtained from adjList, the first 𝑓(Ξ ) rows are shifted so that none of them is repeated. The first 𝑓(Ξ )+1 labelled vertices form a clique and hence the rank π‘Ÿπ‘˜(𝐆) of the adjacency matrix 𝐆 of the 𝑛-vertex 𝐺 which is π‘›βˆ’πœ‚(𝐺) is at least 𝑓(Ξ )+1.

The bound in Theorem 5.2 is reached, for instance, by the threshold graphs 𝐢(𝑓(Ξ )+1) (the complete graph) and by 𝐢(𝑓(Ξ )+1,𝑓(Ξ )).

Theorem 5.3. Let 𝐺 be a threshold graph on 𝑛 vertices, with Durfee square size 𝑓(Ξ ) and nullity πœ‚(𝐺). If 𝑛>2𝑓(Ξ ), then 𝐺 has duplicate vertices.

Proof. The last π‘›βˆ’π‘“(Ξ ) rows of 𝐹(Ξ ) are not affected by the introduction of the zero diagonal when constructing 𝐆 as in Theorem 4.1. Hence, duplicates may only occur among the last π‘›βˆ’π‘“(Ξ ) labelled vertices. If 𝐺 were to have no duplicate vertices, then the last π‘›βˆ’π‘“(Ξ ) rows of 𝐆 need to be all different. Since the 𝑓(Ξ )th row is 𝑓(Ξ ) long, then, by a form of the pigeon-hole principle, the largest number 𝑛 of vertices possible for the graph to have no duplicates is 2𝑓(Ξ ). Therefore if 𝑛>2𝑓(Ξ ), 𝐺 has at least one pair of duplicate vertices.

A threshold graph may have duplicate vertices even if 𝑛<2𝑓(Ξ ). We note again that a kernel eigenvector corresponding to duplicate vertices has only two nonzero entries. This prompts the question: can a kernel eigenvector of the threshold graph have more that two nonzero entries? The answer is in the negative as we will now see.

Theorem 5.4. The nullity πœ‚(𝐺) of a threshold graph 𝐺 is the number of vertices removed to obtain a canonical graph.

Proof. Let 𝐻 be the canonical graph obtained from 𝐺 by removing all the duplicate vertices. Let us say that the number of vertices removed is 𝑑. Since the reflection in the first column of the adjacency matrix 𝐇 of 𝐻 is in row echelon form, then the rows of 𝐇 after the 𝑓(Ξ )π‘‘β„Ž is in strict β€œstepwise” form. Hence, the columns of 𝐇 are linearly independent. Now if the 𝑑 vertices are added to 𝐻 in turn to obtain 𝐺 again, then the nullity increases by one at each stage, contributing to the nullspace of the graph obtained, a kernel eigenvector (with exactly two nonzero entries) while preserving the existing ones. We deduce that there are only 𝑑 linear combinations among the rows of 𝐆 arising from the repeated rows in the last π‘›βˆ’π‘“(Ξ ) rows. Therefore, the nullity of 𝐺 is 𝑑. Moreover, a kernel eigenvector cannot have more than two nonzero entries.

In the proof of Theorem 5.4, the following result becomes evident.

Corollary 5.5. If a threshold graph is singular, then no kernel eigenvector has more than two nonzero entries.

Note that any repeated rows in the first 𝑓(Ξ ) rows of 𝐹(Ξ ) give coduplicates. Also 𝑓(Ξ ) is the degree of a vertex in the first part of the equitable partition of the threshold graph defined by 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) for Lab1. For π΄π‘Ÿ, this corresponds to the ⌈(π‘Ÿ+1)/2βŒ‰th degree in the monotonic nonincreasing sequence of distinct degrees (the ⌈(π‘Ÿ+1)/2βŒ‰th vertex for labeling Lab2).

That an antiregular graph has exactly one pair of either duplicates or coduplicates follows from its construction.

Theorem 5.6. (i)An antiregular graph 𝐴2π‘˜βˆ’1 on an odd number of vertices has a duplicate vertex. (ii)An antiregular graph 𝐴2π‘˜ on an even number of vertices has a coduplicate vertex.

Proof. The graph π΄π‘Ÿ is 𝐢(2,1,1,…,1). Therefore if π‘Ÿ is even, it has a clique of two and hence a pair of coduplicate vertices. On the other hand, if π‘Ÿ is odd, then it has a coclique of two, producing a pair of duplicate vertices.

To obtain the number of duplicate and coduplicate vertices in a threshold graph, we count the number of vertices to be removed from 𝐺 and 𝐺𝐢, respectively, to obtain a canonical graph.

Theorem 5.7. A threshold graph with nondegenerate representation 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ), where π‘Ÿ is even, has(i)βˆ‘π‘Ÿ/2π‘˜=1(π‘Ž2π‘˜βˆ’1βˆ’1) duplicate vertices,(ii)βˆ‘π‘Ÿ/2π‘˜=1(π‘Ž2π‘˜βˆ’1) coduplicate vertices.For odd π‘Ÿ, 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) has(i)βˆ‘(π‘Ÿβˆ’1)/2π‘˜=1(π‘Ž2π‘˜βˆ’1) duplicate vertices,(ii)βˆ‘(π‘Ÿ+1)/2π‘˜=1(π‘Ž2π‘˜βˆ’1βˆ’1) coduplicate vertices.

6. Minimal Configurations

Most of the information to determine the grounds for a labelled graph 𝐺 to be singular is encoded in the nullspace ker(𝐆) of its adjacency matrix 𝐆 (i.e., in ker(𝐆)∢={π±βˆΆπ†π±=𝟎}). The support of a kernel eigenvector 𝐱 in ker(𝐆) is the set of vertices corresponding to the nonzero entries. These vertices induce a subgraph termed the core of 𝐺 with respect to 𝐱. Therefore a core of 𝐺 with respect to 𝐱 is a core graph in its own right. The size of the support is said to be the core order [19].

Definition 6.1 (see [19]). Let 𝐹 be a core graph on at least two vertices, with nullity 𝑠β‰₯1 and a kernel eigenvector 𝐱𝐹 having no zero entries. If a graph 𝑁, of nullity one, having 𝐱𝐹 as the nonzero part of the kernel eigenvector, is obtained by adding π‘ βˆ’1 independent vertices, whose neighbours are vertices of 𝐹, then 𝑁 is said to be a minimal configuration (MC) with core (𝐹,𝐱𝐹).

Hence, an MC with core (𝐹,𝐱𝐹) is a connected singular graph of nullity one having a minimal number of vertices and edges for the core 𝐹, satisfying 𝐅𝐱𝐅=𝟎. The MCs may be considered as the β€œatoms” of a singular graph [19, 20]. The smallest MC is 𝑃3 corresponding to a pair of duplicates. For core order three, the only MC is 𝑃3. The number of MCs increases fast for higher core order (see e.g., [21]). Figure 7 shows two graphs, (a) 𝑃𝐢5, the only MC with core 𝐢4 and (b) a nut graph of order seven [1].

fig7
Figure 7: Two minimal configurations: 𝑃𝐢5 and a nut graph.

A basis for the nullspace ker(𝐆) of the adjacency matrix 𝐆 of a graph 𝐺 of nullity πœ‚>1 can take different forms. We choose a minimal basis 𝐡min for the nullspace of 𝐆, that is, a basis having a minimal total number of nonzero entries in its vectors [19, 22].

Such a minimal basis for ker(𝐆) has the property that the corresponding monotonic non-decreasing sequence of core orders (termed the core order sequence) is unique and lexicographically placed first in a list of bases for ker(𝐆), also ordered according to the nonincreasing core orders. Moreover, for all 𝑖, the 𝑖th entry of the core order sequence for 𝐡min, does not exceed the 𝑖th entry of any other core order sequence of the graph. We say that the vectors in 𝐡min define a fundamental system of cores of 𝐺, consisting of a collection of cores of minimal core order corresponding to a basis of linearly independent nullspace vectors [23]. The significance of MCs can be gauged from the next result.

Theorem 6.2 (see [19, 20]). Let 𝐻 be a singular graph of nullity πœ‚. There exist πœ‚ MCs which are subgraphs of 𝐻 whose core vertices are associated with the nonzero entries of the πœ‚ vectors in a minimal basis of the nullspace of 𝐇.

To give an example supporting Theorem 6.2, Figure 8 shows a six-vertex graph of nullity two and two MCs corresponding to a fundamental system of cores found as subgraphs.

fig8
Figure 8: A graph of nullity two with two MCs as subgraphs.

From Theorem 5.4, the following result follows immediately.

Corollary 6.3. The only MC found in a threshold graph as a subgraph is 𝑃3.

Corollary 6.3 has been generalized to cographs in [24]; that is, in cographs, only 𝑃3 (corresponding to duplicate vertices) may be found as an MC corresponding to a vector in 𝐡min. Therefore it is sufficient to have just 𝑃4 as a forbidden subgraph for a graph to have only core order two contributing to the nullity.

Theorem 6.4. All MCs with core order at least three have 𝑃4 as an induced subgraph.

Proof. Suppose an MC is 𝑃4-free. Then, it is a cograph. Therefore, the only MC to contribute to the nullity is 𝑃3 of core order two. We deduce that all other MCs, which have core order at least three, are not cographs.

Since 𝑃4 is self-complementary, it follows that the complement of an MC with core order at least three also has 𝑃4 as an induced subgraph. Figures 8(b) and 8(c) show 𝑃4 as an induced subgraph (dotted edges) of the MC 𝑃𝐢5.

The second largest eigenvalue of 𝑃4 is the golden section √𝜎∢=(5βˆ’1)/2. By interlacing, we obtain the following result.

Theorem 6.5. The second largest eigenvalue of an MC≠𝑃3 is bounded below by 𝜎.

The only MC for which the bound is known to be strict is the seven-vertex nut graph of Figure 7.

7. The Main Characteristic Polynomial

The main eigenvalues of a graph 𝐺 are closely related to the number of walks in 𝐺. The product of those factors of the minimum polynomial of 𝐆, corresponding to the main eigenvalues only, has interesting properties.

Definition 7.1. The polynomial βˆπ‘€(𝐺,π‘₯)∢=𝑝𝑖=1(π‘₯βˆ’πœ‡π‘–) whose roots are the main eigenvalues of the adjacency matrix of a graph 𝐺 is termed the main characteristic polynomial.

For a proof of the following result, see [25], for instance.

Lemma 7.2 (see [25], rowmain). The main characteristic polynomial 𝑀(𝐺,π‘₯)=π‘₯π‘βˆ’π‘0π‘₯π‘βˆ’1βˆ’π‘1π‘₯π‘βˆ’2βˆ’β‹―βˆ’π‘π‘βˆ’2π‘₯βˆ’π‘π‘βˆ’1 has integer coefficients 𝑐𝑖, for all 𝑖, 0β‰€π‘–β‰€π‘βˆ’1.

7.1. The Main Eigenvalues of Antiregular Graphs

Recall that π΄π‘Ÿ has exactly one pair of either duplicates or coduplicates.

Theorem 7.3. All eigenvalues of π΄π‘Ÿ other than 0 or βˆ’1 are main.

Proof. Let Prop(π‘Ÿ) be all eigenvalues of π΄π‘Ÿ, other than 0 or βˆ’1, are main. We prove Prop(π‘Ÿ) by induction on π‘Ÿ.(i)Prop(2) refers to 𝐾2 whose only nonmain eigenvalue is βˆ’1. Prop(3) refers to 𝑃3 whose only nonmain eigenvalue is 0.
This establishes the base cases.
(ii)Assume that Prop(π‘Ÿ) is true for all π‘Ÿβ‰€π‘˜. Therefore for a nonmain eigenvalue πœ† other than 0 or βˆ’1, π€π‘Ÿπ±=πœ†π± implies 𝐱=0 for π‘Ÿβ‰€π‘˜.(iii)Consider π΄π‘˜+1 and let π€π‘˜+1 be its adjacency matrix.For the case when π‘˜+1 is odd and π΄π‘˜+1 is connected, let π€π‘˜+1𝐱=πœ†π± for an eigenvalue πœ† and 𝐱=(π‘₯1,π‘₯2,…,π‘₯π‘˜+1)β‰ 0. It follows that, for 1β‰€π‘žβ‰€π‘“(Ξ ), βˆ‘π‘˜+2βˆ’π‘žπ‘–=1π‘₯𝑖=(1+πœ†)π‘₯π‘ž and, for 𝑓(Ξ )+1β‰€π‘žβ‰€π‘˜+1, βˆ‘π‘˜+2βˆ’π‘žπ‘–=1π‘₯𝑖=(πœ†)π‘₯π‘ž. Similar equations are obtained for the case when π‘˜+1 is even.
The eigenvalue πœ† is nonmain if and only if 𝐣𝑑𝐱=0, whence πœ†=βˆ’1 or πœ†=0 or π‘₯1=π‘₯2=0.
If 𝑣 (labelled 1) is the dominating vertex of π΄π‘˜+1, then, by Proposition 4.5, π΄π‘˜+1βˆ’π‘£=𝐾1Μ‡βˆͺπ΄π‘˜βˆ’1.
If π‘₯1=π‘₯2=0, then 𝐱 restricted to π΄π‘˜βˆ’1 is an eigenvector for the same eigenvalue πœ†. Therefore, by the induction hypothesis 𝐱=0. Hence, πœ†=βˆ’1 or πœ†=0. The result follows by induction on π‘Ÿ.

7.2. The Main Eigenvalues of Threshold Graphs

By Theorem 7.3, all eigenvalues of π΄π‘Ÿ that are not 0 or βˆ’1 are main. We show that this is still the case for a threshold graph 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) having π‘Ž1β‰₯2 and π‘Žπ‘–β‰₯1 for 2β‰€π‘–β‰€π‘Ÿ obtained from the degenerate form π΄π‘Ÿ=𝐢(1,1,…,1) by adding duplicates and/or coduplicates.

Lemma 7.4. A graph has the same number of main eigenvalues as its complement.

Proof. Let 𝐆𝐂 be the adjacency matrix of the complement of a graph 𝐺 and 𝐉 the matrix with each entry equal to one. Then, 𝐆+𝐆𝐂=π‰βˆ’πˆ. Now πœ† is a nonmain eigenvalue of 𝐺 if and only if 𝐉𝐱=𝟎. Hence, 𝐆 and 𝐆𝐂 share the same eigenvectors only for nonmain eigenvalues.

Theorem 7.5. Let 𝐺 be a threshold graph. All eigenvalues, other than 0 or βˆ’1, are main.

Proof. Let 𝐺 be 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ),π‘Ž1β‰₯2,π‘Žπ‘–β‰₯1 for 2β‰€π‘–β‰€π‘Ÿ. Let the proposition Prop(π‘Ÿ) be all eigenvalues of 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) other than 0 or βˆ’1, are main. We prove Prop(π‘Ÿ) by induction on π‘Ÿ.(i)If 𝐺=𝐢(π‘Ž1,π‘Ž2),π‘Ž1β‰₯2,π‘Ž2β‰₯1, then 𝐺 is not regular. Hence, the number of main eigenvalues is at least two. The other distinct eigenvalues, 0 and/or βˆ’1, are nonmain. By Theorem 5.7, 𝐺 has at least π‘›βˆ’2 nonmain eigenvalues equal to 0 or βˆ’1. Thus, the number of main eigenvalues of 𝐺 is two. This establishes the base case, namely, Prop(2).(ii)The induction hypothesis is as follows: assume that Prop(π‘˜) is true.(iii)We show that this is also true for a nondegenerate 𝐻=𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘˜+1).The complement 𝐻 of 𝐻 is 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘˜)Μ‡βˆͺπ‘Žπ‘˜+1𝐾1. By Lemma 7.4, 𝐻 and 𝐻 have the same number of main eigenvalues. One of the π‘Žπ‘˜+1 isolated vertices in 𝐻 contributes to the number of main eigenvalues. By the induction hypothesis, 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘˜) has π‘˜ main eigenvalues and βˆ‘π‘˜π‘–=1(π‘Žπ‘–βˆ’1) nonmain eigenvalues. Hence, 𝐻 has π‘˜+1 main eigenvalues. The result follows by induction on π‘Ÿ.

We deduce immediately a spectral property of a threshold graph and its underlying antiregular graph.

Corollary 7.6. The nondegenerate threshold graph 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) and its underlying π΄π‘Ÿ have π‘Ÿ and π‘Ÿβˆ’1 main eigenvalues, respectively.

An equitable partition Π∢=𝒱1,𝒱2,…,π’±π‘Ÿ of the vertex set of a graph satisfies 𝐆𝐗=𝐗𝐐, where X is the π‘›Γ—π‘Ÿ indicator matrix whose 𝑖th column is the characteristic 0-1-vector associated with the 𝑖th part, containing |𝒱𝑖| entries equal to 1. The matrix 𝐐 turns out to be the adjacency matrix of the quotient graph 𝐺/Ξ  (also known as divisor).

Lemma 7.7. The main part of the spectrum of 𝐆 is included in the spectrum of 𝐐.

Proof. Let πœ† be a main eigenvalue of 𝐺. Then, 𝐆𝐱=πœ†π±, where 𝐣𝑑𝐱≠0. Since 𝐆𝐗=𝐗𝐐, πœ†π±π‘‘π—=𝐱𝑑𝐆𝐗=(𝐱𝑑𝐗)𝐐 so that πœ†(𝐗𝑑𝐱)=𝐐(𝐗𝑑𝐱). Thus, the eigenvalue πœ† of 𝐺 is also an eigenvalue of 𝐐, provided that 𝐗𝑑𝐱≠0. Indeed this is the case when πœ† is a main eigenvalue, since 𝐱𝑑⋅𝐗,𝐣=𝐱.𝐣≠0. Thus, the main part of the spectrum of 𝐺 is contained in the spectrum of 𝐐.

We now show that the main part of the spectrum of 𝐺=𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) is precisely the spectrum of 𝐐. Consider the equitable vertex partition Ξ  for 𝐺=𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) as outlined in Section 3.2.

Theorem 7.8. Let the threshold graph 𝐺=𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) have πœ‚ duplicates, πœ‚ coduplicates, and an equitable partition Ξ  corresponding to the parts {π‘Žπ‘–}. Let 𝐐 be the adjacency matrix of the quotient graph 𝐺/Ξ . Then, πœ™(𝐺,πœ†)=πœ†πœ‚(1+πœ†)πœ‚πœ™(𝐐,πœ†), where πœ™(𝐐) is the main characteristic polynomial 𝑀(𝐺,πœ†) of 𝐺.

Proof. The vertex labelling Lab1 is used. Let the vertices be labelled in order starting from those corresponding to π‘Ž1, followed by those for π‘Ž2 and so on. If X is the π‘›Γ—π‘Ÿ indicator matrix whose 𝑖th column is the characteristic 0-1-vector associated with π‘Žπ‘– containing exactly π‘Žπ‘– nonzero entries (each equal to 1), then 𝐆𝐗=𝐗𝐐, where 𝐐 is π‘ŸΓ—π‘Ÿ. Now, by Theorem 7.5, in a threshold graph, 0 and βˆ’1 are the only nonmain eigenvalues and these correspond to duplicates and coduplicates, respectively. Therefore, the number of main eigenvalues of 𝐆 is exactly π‘Ÿ. Since the main spectrum of 𝐆 is contained in the spectrum of 𝐐 and 𝐐 is π‘ŸΓ—π‘Ÿ, then the roots of πœ™(𝐐) are the main eigenvalues of 𝐺.

We give an example to clarify the procedure. Consider the threshold graph 𝐺=𝐢(2,2,1,2,1,2) (Lab1), of Figure 3. We use the adjacency matrix 𝐆 and indicator matrix 𝐗, indexed according to Lab2: βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ π†=0111111111101111111111011111101110111110111011110011110111001111110000111111000011110000001100000000,𝐗=100000100000010000010000001000001000000100000100000010000001.(7.1) The rows of 𝐐 are the distinct rows of 𝐆𝐗. Therefore, βŽ›βŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽ π=122211212210221200222000220000200000.(7.2) Its spectrum is 7.16, 0.892, 0.448, βˆ’1.40, βˆ’1.59, βˆ’2.50, which is precisely the main part of the spectrum of 𝐺.

For β„“β‰₯0, the entries of 𝐆ℓ𝐣 give the number of walks of length β„“ from each vertex 𝑣 of 𝐺. The π‘›Γ—π‘˜ matrix whose β„“th column is π†β„“βˆ’1𝐣 is denoted by π–π‘˜. The dimension of the subspace ColSp(π–π‘˜) generated by the columns of π–π‘˜ is the rank of π–π‘˜.

Theorem 7.9 (see [26]). For a graph with 𝑝 main eigenvalues, the rank, dim(ColS𝑝(π–π‘˜)), of the π‘›Γ—π‘˜ matrix π–π‘˜=(𝐣,𝐆𝐣,𝐆2𝐣,…,π†π‘˜βˆ’1𝐣) is 𝑝, for π‘˜β‰₯𝑝.

The columns 𝐣,𝐆𝐣,𝐆2𝐣,β‹―,π†π‘βˆ’1𝐣 are a maximal set of linearly independent vectors in ColSp(π–π‘˜). Thus, the first 𝑝 columns provide all the information on the number of walks from each vertex of any length [27].

Definition 7.10. The matrix 𝐖𝑝=(𝐣,𝐆𝐣,𝐆2𝐣,…,π†π‘βˆ’1𝐣) of rank 𝑝 is said to be the walk matrix 𝐖.

Note that 𝐖 has the least number of columns for a walk matrix π–π‘˜ to reach the maximum rank possible which is 𝑝. From Corollary 7.6, 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) has π‘Ÿ main eigenvalues.

Theorem 7.11. The rank of the walk matrix of 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) is π‘Ÿ.

The number of walks of length π‘˜ can be expressed in terms of the main eigenvalues [28, page 46].

Theorem 7.12. The number π‘€π‘˜ of walks of length π‘˜β€‰β€‰starting from any vertex of 𝐺 is given by π‘€π‘˜=𝑝𝑖=1π‘ξ…žπ‘–πœ‡π‘˜π‘–,(7.3) where π‘ξ…žπ‘–βˆˆβ„β§΅{0} is independent of π‘˜ for each 𝑖 and πœ‡1,πœ‡2,…,πœ‡π‘ are the main eigenvalues of 𝐆.

Since 0 is never a main eigenvalue of 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ), it follows that all the main eigenvalues of 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) contribute to the number of walks.

7.3. Cases of Reducible Main Polynomial

By Theorem 7.3, only one eigenvalue of π΄π‘Ÿ is not main. Recall that the minimal equitable vertex partition of 𝐺=𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) satisfies 𝐆𝐗=𝐗𝐐, where 𝐐 is the adjacency matrix of the quotient graph 𝐺/Ξ  and πœ™(𝐐,πœ†)=𝑀(𝐺,πœ†), the main characteristic polynomial 𝑀(𝐺,πœ†) of 𝐺.

We note that for many threshold graphs πœ™(𝐐,πœ†) is irreducible over the integers. For example the only eigenvalue of 𝐴8=𝐢(1,1,1,1,1,1,1,1) (in degenerate form) which is not main is βˆ’1 and 𝑀(𝐴8,π‘₯)=(1βˆ’7π‘₯+9π‘₯2+15π‘₯3βˆ’13π‘₯4βˆ’15π‘₯5βˆ’π‘₯6+π‘₯7), which is irreducible.

Now we add vertices to the degenerate form 𝐴8=𝐢(1,1,1,1,1,1,1,1). If we add a vertex to the first part, to obtain 𝐺1=𝐢(2,1,1,1,1,1,1,1), a negative eigenvalue (not βˆ’1) and 0 appear. The eigenvalue βˆ’1 is lost and 𝑀(𝐺1,π‘₯)=(2βˆ’12π‘₯+6π‘₯2+40π‘₯3βˆ’40π‘₯5βˆ’20π‘₯6+π‘₯8). When a vertex is added to the third part to obtain 𝐺3=𝐢(2,2,1,1,1,1,1), the eigenvalue βˆ’1 is retained while the zero eigenvalue appears and 𝑀(𝐺3,π‘₯)=(2βˆ’12π‘₯+12π‘₯2+22π‘₯3βˆ’16π‘₯4βˆ’18π‘₯5βˆ’π‘₯6+π‘₯7). In both these latter two cases πœ™(𝐐,πœ†) is irreducible over the integers. Now when a vertex is added to the seventh part to obtain 𝐺7=𝐢(2,1,1,1,1,2,1), the eigenvalue βˆ’1 is retained while the zero eigenvalue appears. In this case, however, 𝑀(𝐺7,π‘₯)=(π‘₯2+2π‘₯βˆ’1)(π‘₯5βˆ’3π‘₯4βˆ’9π‘₯2+3π‘₯3+8π‘₯βˆ’2), and therefore it is reducible over the integers.

This is also the case for some instances of the threshold graphs 𝐢(𝑑,1,𝑑) when the cubic polynomial πœ™(𝐐,πœ†) has an integer as a root and therefore is reducible. The divisor 𝐐 is ξ‚€π‘‘βˆ’10𝑑00𝑑𝑑1π‘‘βˆ’1 with characteristic polynomial πœ™(𝐐,πœ†)=βˆ’π‘‘+𝑑𝑑+πœ†βˆ’π‘‘πœ†βˆ’2π‘‘πœ†+2πœ†2βˆ’π‘‘πœ†2βˆ’π‘‘πœ†2+πœ†3.

If πœ† is 0, 2 or 3, there are no integral values of 𝑑 and 𝑑 satisfying the polynomial πœ™(𝐐,πœ†). If πœ†=1, the graph either for 𝑑=3 and 𝑑=8 or for 𝑑=4 and 𝑑=6 satisfies it. Also for πœ†=βˆ’2 either the graph for 𝑑=3 and 𝑑=5, or 𝑑=4 and 𝑑=3, or 𝑑=6 and 𝑑=2 satisfies it, while for πœ†=βˆ’3, the graph for 𝑑=7 and 𝑑=40 satisfies it.

8. Sign Pattern of the Spectrum of a Threshold Graph

We conclude with a note on the distribution of the eigenvalues of a threshold graph. In [29] it was remarked that an antiregular graph has a bipartite character, that is, the number π‘Ÿβˆ’ of negative eigenvalues is equal to the number π‘Ÿ+ of positive ones. We denote the number of zero eigenvalues by πœ‚.

8.1. The Spectrum of π΄π‘Ÿ

For 𝑛β‰₯4, π΄π‘Ÿ is not bipartite. Therefore, βˆ’πœ†minβ‰ πœ†max. The proof of the next result is by induction on the order of the antiregular graph. We will need the following evident fact.

Lemma 8.1. To transform π΄π‘Ÿ to π΄π‘Ÿ+1 (according to the labelling (Lab2) of the stepwise adjacency matrix),(i)a vertex duplicate to the ⌈(π‘Ÿ+1)/2βŒ‰π‘‘β„Ž is added for even π‘Ÿ,(ii)a vertex coduplicate to the ⌈(π‘Ÿ+1)/2βŒ‰π‘‘β„Ž is added for odd π‘Ÿ.

Theorem 8.2. π‘Ÿ+=π‘Ÿβˆ’ for π΄π‘Ÿ.

Proof. The proof is by induction on π‘Ÿ.
The spectra of the three smallest antiregular graphs, Sp(𝐴1)={0}, Sp(𝐴2)={βˆ’1,1}, and Sp(𝐴3√)={βˆ’βˆš2,0,2}, establish the base cases.
Assume that the theorem is true for π΄π‘˜.
We prove it true for π΄π‘˜+1.
If π΄π‘˜ is singular, then it has a duplicate vertex and π‘˜ is odd. By the induction hypothesis π‘Ÿ+=π‘Ÿβˆ’.
If, on the other hand, π΄π‘˜ is nonsingular, then π΄π‘˜ has a coduplicate vertex and π‘˜ is even. Again the nonzero eigenvalues satisfy π‘Ÿ+=π‘Ÿβˆ’.
We apply Lemma 8.1, using Lab2. For odd π‘˜, if a vertex 𝑀, coduplicate to the ⌈(π‘Ÿ+1)/2βŒ‰th vertex, is added to π΄π‘˜, then only one of the duplicate vertices of π΄π‘˜ will have 𝑀 as a neighbour in π΄π‘˜+1. The zero eigenvalue of π΄π‘˜ vanishes and the eigenvalue βˆ’1 is introduced for π΄π‘˜+1. By the Perron Frobenius theorem adding edges to a graph (π΄π‘˜Μ‡βˆͺ𝐾1) increases the maximum eigenvalue. Therefore, by interlacing, the number of positive eigenvalues increases by one. Since the new coduplicate vertex 𝑀 contributes the new eigenvalue βˆ’1 to the spectrum, it follows that π‘Ÿ+=π‘Ÿβˆ’ will be satisfied in π΄π‘˜+1. By interlacing, adding a duplicate vertex to any graph retains the number of positive and negative eigenvalues and adds 0 to the spectrum. For even π‘˜, if a vertex 𝑀, duplicate to the ⌈(π‘Ÿ+1)/2βŒ‰th vertex, is added, then a duplicate vertex is added to the graph, retaining π‘Ÿ+=π‘Ÿβˆ’.
The result follows by induction on 𝑛.

8.2. The Spectrum of a Threshold Graph

In this section, we shall represent the antiregular graph π΄π‘Ÿ by the degenerate form 𝐢(1,1,…,1). As in Section 4, any part can be expanded to produce a threshold graph 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ). We need the following evident facts regarding the effect on the distribution of the spectrum of the adjacency matrix when a vertex is added.

Lemma 8.3. If on adding a vertex to a graph (i) the multiplicity of an eigenvalue πœ†0 of the adjacency matrix increases, then, by interlacing, the number π‘›βˆ’(πœ†0) of eigenvalues less than πœ†0 and the number 𝑛+(πœ†0) greater than πœ†0 remain the same; (ii) the multiplicity of an eigenvalue πœ†0 of the adjacency matrix decreases, then by interlacing, each of the numbers 𝑛+(πœ†0) and π‘›βˆ’(πœ†0) increases by one.

We shall write 𝑛+ for 𝑛+(0) and π‘›βˆ’ for π‘›βˆ’(0).

First we see an application of Lemma 8.3(i) using Lab1. For even π‘Ÿ, if one of the even indexed π‘Žπ‘–, for 𝑖β‰₯2, of 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) is increased, then a coduplicate of a vertex is added. This forces πœ‚ and 𝑛+ to remain unchanged while each of π‘›βˆ’ and the multiplicity π‘š(βˆ’1) of the eigenvalue βˆ’1 increases by one. If the odd indexed π‘Žπ‘–, for some 𝑖β‰₯1, is increased, then a duplicate of a vertex is added forcing 𝑛+ and π‘›βˆ’ to remain unchanged.

Similarly, for odd π‘Ÿ, if the even indexed π‘Žπ‘–, for some 𝑖β‰₯2, is increased, then a duplicate of a vertex is added. This forces π‘›βˆ’ and 𝑛+ to remain unchanged while πœ‚ increases by one. If the odd indexed π‘Žπ‘–, for some 𝑖β‰₯3, is increased, then a coduplicate of a vertex is added forcing 𝑛+ and πœ‚ to remain unchanged.

The case for even π‘Ÿ and π‘Ž1>1 is the same as for odd π‘Ÿ with π‘Ž1=1 (Lab1). Taking 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) for odd π‘Ÿ with π‘Ž1=1 and expanding to 𝐢(π‘Ž1,π‘Ž2,π‘Ž3,…,π‘Žπ‘Ÿ) with π‘Ž1>1 gives the unique case where πœ‚ decreases by one and π‘š(βˆ’1) increases by one. Since πœ‚ decreases by one, by Lemma 8.3(ii), each of 𝑛+ and π‘›βˆ’ increases by one, the latter corresponding to the increase in the multiplicity of the eigenvalue βˆ’1. We have proved the following result.

Theorem 8.4. If the threshold graph 𝐢(π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ) is transformed to another threshold graph by increasing exactly one of the π‘Žπ‘–s by one, then ⎧βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽ©π‘–π‘“π‘Žπ‘‘π‘’π‘π‘™π‘–π‘π‘Žπ‘‘π‘’π‘–π‘ π‘Žπ‘‘π‘‘π‘’π‘‘,π‘‘β„Žπ‘’π‘›π‘›βˆ’π‘Žπ‘›π‘‘π‘›+π‘Žπ‘Ÿπ‘’π‘’π‘›π‘β„Žπ‘Žπ‘›π‘”π‘’π‘‘π‘Žπ‘›π‘‘πœ‚π‘–π‘›π‘π‘Ÿπ‘’π‘Žπ‘ π‘’π‘ ;π‘–π‘“π‘Žπ‘π‘œπ‘‘π‘’π‘π‘™π‘–π‘π‘Žπ‘‘π‘’π‘–π‘ π‘Žπ‘‘π‘‘π‘’π‘‘,π‘Žπ‘›π‘‘π‘–π‘“π‘Ÿπ‘–π‘ π‘’π‘£π‘’π‘›π‘‘β„Žπ‘’π‘›πœ‚π‘Žπ‘›π‘‘π‘›+π‘Žπ‘Ÿπ‘’π‘’π‘›π‘β„Žπ‘Žπ‘›π‘”π‘’π‘‘π‘œπ‘Ÿπ‘–π‘“π‘Ÿπ‘–π‘ π‘œπ‘‘π‘‘π‘Žπ‘›π‘‘π‘Žπ‘–β‰₯3π‘œπ‘Ÿπ‘–π‘“π‘Ÿπ‘–π‘ π‘œπ‘‘π‘‘π‘Žπ‘›π‘‘π‘Ž1>1,π‘Žπ‘›π‘‘π‘›βˆ’π‘–π‘›π‘π‘Ÿπ‘’π‘Žπ‘ π‘’π‘ ;π‘–π‘“π‘Žπ‘π‘œπ‘‘π‘’π‘π‘™π‘–π‘π‘Žπ‘‘π‘’π‘–π‘ π‘Žπ‘‘π‘‘π‘’π‘‘,π‘Žπ‘›π‘‘π‘–π‘“π‘Ÿπ‘–π‘ π‘œπ‘‘π‘‘π‘Žπ‘›π‘‘π‘Ž1=1π‘‘β„Žπ‘’π‘›π‘›βˆ’π‘Žπ‘›π‘‘π‘›+π‘–π‘›π‘π‘Ÿπ‘’π‘Žπ‘ π‘’π‘Žπ‘›π‘‘πœ‚π‘‘π‘’π‘π‘Ÿπ‘’π‘Žπ‘ π‘’π‘ .(8.1)

9. Conclusion

The simple graphic appeal of the Ferrers/Young diagram 𝐹(Ξ ), with rows representing the degree sequence of a 𝑛-vertex threshold graph has been instrumental to obtain interesting results on the nullity and structure of the graph. The shape of 𝐹(Ξ ) has been also used to determine the nature of the eigenvalues as main or nonmain.

Let 𝐃 be the diagonal entries whose nonzero entries are the vertex degrees for some labelling of the vertices. Like the adjacency matrix 𝐀, the Laplacian πƒβˆ’π€ also gives a wealth of information about the graph. It is well known that the class of graphs for which the Laplacian spectrum and the conjugate degree sequence πœ‹βˆ— (i.e., the lengths of the columns of 𝐹(Ξ )) coincide is exactly the class of threshold graphs [30, Chapter 10]. The Grone-Merris Conjecture, asserting that the spectrum of the Laplacian matrix of a finite graph is majorized by the conjugate degree sequence of the graph, has been recently proved by Bai [31].

Acknowledgments

This paper was supported by the Research Project Funds MATRP01-01 Graph Spectra and Fullerene Molecular Structures of the University of Malta.

References

  1. I. Sciriha and I. Gutman, β€œNut graphs: maximally extending cores,” Utilitas Mathematica, vol. 54, pp. 257–272, 1998. View at Zentralblatt MATH
  2. F. R. Gantmacher II, The Theory of Matrices, Chelsea, New York, NY, USA, 1960.
  3. V. Chvátal and P. L. Hammer, β€œAggregation of inequalities in integer programming,” in Studies in Integer Programming (Proc. Workshop, Bonn, 1975), P. L. Hammer, E. L. Johnson, B. H. Korte, et al., Eds., Annals of Discrete Mathematics, 1, pp. 145–162, North-Holland, Amsterdam, The Netherlands, 1977. View at Zentralblatt MATH
  4. P. B. Henderson and Y. Zalcstein, β€œA graph-theoretic characterization of the PV class of synchronizing primitives,” SIAM Journal on Computing, vol. 6, no. 1, pp. 88–108, 1977. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, vol. 57 of Annals of Discrete Mathematics, Elsevier Science, Amsterdam, The Netherlands, 2nd edition, 2004.
  6. N. V. R. Mahadev and U. N. Peled, Threshold Graphs and Related Topics, vol. 56 of Annals of Discrete Mathematics, North-Holland, Amsterdam, The Netherlands, 1995.
  7. R. Merris, Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, NY, USA, 2001.
  8. A. Brandstädt, V. B. Le, and J. P. Spinrad, Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1999.
  9. H. A. Jung, β€œOn a class of posets and the corresponding comparability graphs,” Journal of Combinatorial Theory. Series B, vol. 24, no. 2, pp. 125–133, 1978. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. H. Lerchs, β€œOn cliques and kernels,” Tech. Rep., Department of Computer Science, University of Toronto, 1971.
  11. D. Seinsche, β€œOn a property of the class of n-colorable graphs,” Journal of Combinatorial Theory. Series B, vol. 16, pp. 191–193, 1974. View at Publisher Β· View at Google Scholar
  12. D. P. Sumner, β€œDacey graphs,” Journal of the Australian Mathematical Society, vol. 18, pp. 492–502, 1974. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. T. Biyikoglu, S. K. Simi, and Z. Stanić, β€œSome notes on spectra of cographs,” Ars Combinatoria, vol. 100, pp. 421–434, 2011.
  14. D. G. Corneil, Y. Perl, and L. K. Stewart, β€œA linear recognition algorithm for cographs,” SIAM Journal on Computing, vol. 14, no. 4, pp. 926–934, 1985. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  15. E. Ruch and I. Gutman, β€œThe branching extent of graphs,” Journal of Combinatorics, Information & System Sciences, vol. 4, no. 4, pp. 285–295, 1979. View at Zentralblatt MATH
  16. R. A. Brualdi and A. J. Hoffman, β€œOn the spectral radius of (0, 1)-matrices,” Linear Algebra and Its Applications, vol. 65, pp. 133–146, 1985. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  17. R. A. Brualdi and E. S. Solheid, β€œOn the spectral radius of connected graphs,” Institut Mathématique. Publications. Nouvelle Série, vol. 39(53), pp. 45–54, 1986. View at Zentralblatt MATH
  18. Z. Stanić, β€œOn nested split graphs whose second largest eigenvalue is less than 1,” Linear Algebra and Its Applications, vol. 430, no. 8-9, pp. 2200–2211, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  19. I. Sciriha, β€œA characterization of singular graphs,” Electronic Journal of Linear Algebra, vol. 16, pp. 451–462, 2007. View at Zentralblatt MATH
  20. I. Sciriha, β€œOn the rank of graphs,” in Combinatorics, Graph Theory, and Algorithms, Y. Alavi, D. R. Lick, and A. Schwenk, Eds., vol. 2, pp. 769–778, New Issues Press, Kalamazoo, Mich, USA, 1999.
  21. I. Sciriha, β€œOn the construction of graphs of nullity one,” Discrete Mathematics, vol. 181, no. 1–3, pp. 193–211, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  22. I. Sciriha, S. Fiorini, and J. Lauri, β€œMinimal basis for a vector space with an application to singular graphs,” Graph Theory Notes of New York, vol. 31, pp. 21–24, 1996.
  23. I. Sciriha, β€œMaximal core size in singular graphs,” Ars Mathematica Contemporanea, vol. 2, no. 2, pp. 217–229, 2009. View at Zentralblatt MATH
  24. G. F. Royle, β€œThe rank of a cograph,” The Electronic Journal of Combinatorics, vol. 10, no. 1, 2003. View at Zentralblatt MATH
  25. D. Cvetković and M. Petrić, β€œA table of connected graphs on six vertices,” Discrete Mathematics, vol. 50, no. 1, pp. 37–49, 1984. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  26. E. M. Hagos, β€œSome results on graph spectra,” Linear Algebra and Its Applications, vol. 356, pp. 103–111, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  27. I. Sciriha and D. M. Cardoso, β€œNecessary and sufficient conditions for a hamiltonian graph,” Journal of Combinatorial Mathematics and Combinatorial Computing. In press.
  28. D. M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, Johann Ambrosius Barth, Heidelberg, Germany, 3rd edition, 1995.
  29. R. Merris, β€œAntiregular graphs are universal for trees,” Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika, vol. 14, pp. 1–3, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  30. R. B. Bapat, Graphs and Matrice, Hindustan Book Agency, New Delhi, India, 2011.
  31. H. Bai, β€œThe Grone-Merris conjecture,” Transactions of the American Mathematical Society, vol. 363, no. 8, pp. 4463–4474, 2011. View at Publisher Β· View at Google Scholar