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ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 108509, 21 pages
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 © 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.
The antiregular connected graph on vertices is defined as the connected graph whose vertex degrees take the values of 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 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.
A graph of order has a labelled vertex set containing vertices and a set of edges consisting of unordered pairs of the vertices. When a subset of is deleted, the edges incident to are also deleted. The subgraph of is said to be an induced subgraph of . The subgraph of obtained by deleting a particular vertex is simply denoted by . The cycle and the complete graph on vertices are denoted by and , respectively.
The graphs we consider are simple, that is, without loops or multiple edges. We use bold face, say , to denote the 0-1-adjacency matrix of the graph bearing the same name , where the entry of the symmetric matrix is 1 if and 0 otherwise. We note that the graph is determined, up to isomorphism, by . The adjacency matrix of the complement of is , where each entry of is one and is the identity matrix. The degree of a vertex is the number of nonzero entries in the row of .
The disconnected graph with two components and is their disjoint union, denoted by . For , the graph is the disconnected graph with components, where each component is isomorphic to . The join of and is .
For the linear transformation , the real numbers satisfying for some nonzero vector are said to be eigenvalues of and form the spectrum of . They are the solutions of the characteristic polynomial of , defined as the polynomial in . The subspace of that maps to zero under is said to be the nullspace of . A graph is said to be singular of nullity if the dimension of is . The nonzero vectors, , in the nullspace, termed kernel eigenvectors of , satisfy . We note that the multiplicity of the eigenvalue zero is . If there exists a kernel eigenvector of with no zero entries, then is said to be a core graph. The cycle on four vertices is a core graph of nullity two with a kernel eigenvector for the usual labelling of the vertices round the cycle. A core graph of nullity one is said to be a nut graph . 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 , which have an associated eigenvector not orthogonal to (the vector with each entry equal to one) are said to be main. We denote the remaining distinct eigenvalues by , 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 by complementation and disjoint union. Threshold graphs are a subclass of cographs. They were first introduced in 1977 by Chvátal and Hammer in connection with the equivalence between set packing and knapsack problems  and independently, in the same year, by Henderson and Zalcstein for parallel systems in computer programming . 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 of into parts equal to the vertex degrees is needed. The array of boxes , known as a Ferrers/Young diagram for the monotonic nonincreasing sequence consists of rows of boxes as runs successively from 1 to . 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 , the cycle and the two copies of the complete graph on two vertices. It is said to be -, -, and -free.
Equivalently, (ii) if the monotonic nonincreasing degree sequence, , of a graph is represented by the rows of a Ferrers/Young diagram , where the length of the principal square of is and the lengths of the columns of satisfy , then is said to be a threshold graph [7, Lemma 7.23].
If the parts of a threshold graph partition of 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 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 vertices is defined as a threshold graph whose vertex degrees take as many different values as possible, that is, distinct nonnegative integral values.
Definition 1.3. The partition of the vertex set of a graph is said to be an equitable partition if, for all , the number of neighbours in of a vertex in depends only on the choice of and .
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 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 induced.(iii)We show in Section 7 that the spectrum of a connected threshold graph and its underlying antiregular graph show common characteristics. All the eigenvalues other than 0 and −1 are main and each main eigenvalue contributes to the number of walks. Moreover, the spectrum of its quotient graph consists precisely of the main eigenvalues of . The characteristic polynomial of is reducible over the integers (i.e., it has polynomial factors) for certain threshold graphs .(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.
A cograph is the union or the join of subgraphs of the form , where , for all . Therefore, the family of cographs is the smallest class of graphs that includes and is closed under complementation and disjoint union. It is well known that no cograph on at least four vertices has as an induced subgraph . In fact cographs can also be characterized as -free graphs.
Cographs have received much attention since the 1970s. They were discovered independently by many authors including Jung  in 1978, Lerchs  in 1971 and, Seinsche  and Sumner , both in 1974. For a more detailed treatment of cographs, see .
Connected graphs, which are -, -, and -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 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 is , then both and have -induced. Thus there exist connected graphs that are neither free nor have a connected complement.
Recall that . Hence, cographs are also characterized as the smallest class of graphs that includes and is closed under join and disjoint union. On this definition of cographs, the proofs in , 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  and later in . Figure 1 shows the cotree of the cograph . The vertices , , and 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 are vertices of . The cotree is a rooted tree and only the terminal vertices represent the cograph vertices. An interior vertex or of represents the subgraph of induced by its terminal successors. The immediate successors of can be cograph vertices or . Similarly the immediate successors of can be cograph vertices or . Therefore, the interior vertices of on a (oriented) path descending from the root to a terminal vertex of are a sequence of alternating and .
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 .
Lemma 3.1. If a cograph is also a threshold graph, then each interior vertex of has at most one interior vertex as an immediate successor.
Proof. A threshold graph is -free and therefore is a cograph which can be represented by a cotree . Note that cannot be represented as a cotree. In a threshold graph, there are no induced subgraphs isomorphic to or to . Therefore, the configurations in Figure 2(a) representing and 2(b) representing as cotrees are not allowed in the cotree corresponding to a threshold graph . We deduce that the number of interior vertices which are immediate successors of an interior vertex is less than two, as required.
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 (vertex 1a), the graph in Figure 3 is coded as to avoid repetitions of successive joins or unions. Therefore, according to the vertex labelling in Figure 3, is . The cotree represents the threshold graph drawn next to it in a way so as to emphasise the nested split graph structure of , 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.
In , the terminal vertices which are immediate successors of a vertex form a clique (inducing a complete subgraph) whereas those immediately succeeding a vertex form a coclique (inducing a subgraph without edges). A line in joining and , which are circumscribed cliques or cocliques, means that each vertex of is adjacent to each vertex of .
3.2. Minimal Equitable Partition of the Vertex Set
Our labelling of the parts in the equitable partition of the vertices of a connected threshold graph follows the addition of the vertices in the construction in order, namely, 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 for . For a minimal value of , is said to be a nondegenerate equitable partition for the nondegenerate representation . All other equitable partitions of the vertex set are refinements of with a larger number of parts, when an equitable partition and the corresponding representation are said to be degenerate. Unless otherwise stated we will assume that equitable vertex partitions and representations are nondegenerate. In particular .
According to our labelling convention (Lab1) for as in Figure 3, a threshold graph whose cotree has root is connected. If is even, then is associated with a coclique, whereas, for odd, is associated with a clique. It follows that the monotonic nonincreasing vertex degree sequence of will be associated with in that order if is even and in that order if is odd. By convention therefore, for a nondegenerate equitable partition, for and . According to this representation, the graph of Figure 3 has the nondegenerate representation .
3.3. The Binary Code of a Threshold Graph
For the purposes of inputting an -vertex-threshold graph to be processed in a computer program, the graph is encoded as a string of 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 , 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 be the Ferrers/Young diagram (Figure 4) for the nonincreasing degree sequence giving a vertex partition of for an -vertex graph. The largest principal square of boxes in is termed the Durfee square and 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 for .
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 of a threshold graph also produces a particular structure of the Ferrers/Young diagram , shown in Figure 4.
Lemma 3.4 (see ). For a threshold graph, consists of four blocks , , , and its transpose , where is the Durfee square, is the row of of length , and is the array of boxes left after removing the Durfee square from the first rows of .
4. The Structure of Threshold Graphs
An interesting algorithm was presented in  to construct a threshold graph. The adjacency list of the graph, that is the list of neighbours of each vertex, is in fact obtained by filling in the boxes of the row in with consecutive integers starting from 1, but skipping . By Lemma 3.3, 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 . We note that this gives us the second labelling, Lab2, in order of the nonincreasing degree sequence and therefore different from Lab1 used for .
Theorem 4.1. The adjacency matrix of a threshold graph is obtained from its Ferrers/Young diagram , representing the degree sequence of a -vertex graph, as follows. The box is inserted in each 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 array of boxes is obtained. Each of the remaining empty boxes is filled with zero. The array of 0-1-numbers obtained is the adjacency matrix .
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 row in is filled with to obtain , then the adjacency list adjList of the graph is just a rearrangement of the entries of since, by Definition 1.1, . Due to the shape of the nonzero part, the adjacency matrix is said to have “a stepwise” form [16, 17].
4.1. The Antiregular Graph
The antiregular graph may be considered to be the smallest threshold graph for an equitable vertex partition having a given number of parts.
Definition 4.2. An antiregular graph on vertices is a graph whose vertex degrees take the values of distinct (nonnegative) integers.
We shall use the -vertex connected antiregular graph with the largest number of parts in its equitable partition, having degenerate representation using Lab1. Any part can be expanded to produce a threshold graph , taking care to preserve the nested split structure. The connected antiregular graph with degenerate equitable partition into parts is adopted as the underlying graph of a connected threshold graph for an equitable vertex partition with parts.
Lemma 4.3. An induced subgraph of is , where for .
Proof. When for at least one value of , to produce , vertices are deleted from the part of size in the equitable partition of . This procedure produces an induced subgraph at each stage and it is repeated until is reached for each .
The threshold graph having parts, where each part is of size 1, is the degenerate form of . Its nondegenerate form, consistent with the cotree representations of threshold graphs, is having 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 , having parts, with degenerate representation , having parts, is an induced subgraph of where for and .
Proposition 4.5. Let be the dominating vertex of . Then, (i) is and (ii).
Figures 5 and 6, respectively, show the threshold graphs with underlying and , their complements, and the -deleted subgraphs when is the only dominating vertex. The corresponding representations of and are and , respectively.
Proposition 4.6. The binary codes for the connected antiregular graphs and are, respectively, the -string and the -string with alternating 0 and 1 entries.
Since the binary code follows the construction of algorithmically, we have the following.
Corollary 4.7. The construction of connected antiregular graphs is as follows: for :, .
4.2. The Complement of a Threshold Graph
The complement of a connected threshold graph is disconnected and is denoted by (see Figure 3). The following result is deduced from the construction of the complement.
Proposition 4.8 (see ). The cotree of the complement of is obtained from by changing the interior vertices from to and viceversa.
Corollary 4.9. The complement of the connected threshold graph , is the disconnected threshold graph isomorphic to .
Proof. Since is connected, its cotree has as a root. Therefore, by Proposition 4.8, the cotree has as a root, and therefore it has coclique .
Proposition 4.10. The binary string coding of the threshold graph , with the underlying graph , is the -string of 0 and 1 entries. (The superscripts denote repetition; denotes the substring 111 with 1 repeated times).
Similarly the binary string coding of the threshold graph , with underlying graph , is the -string .
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 and , the and rows differ only in the and entries. It follows that both duplicates and coduplicates produce the eigenvector with only two nonzero entries, namely, 1 and −1, at positions corresponding to the pair of vertices, with corresponding eigenvalue 0 and −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 “stepwise” 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 of the adjacency matrix of a threshold graph is .
Proof. When the adjacency matrix is obtained from adjList, the first rows are shifted so that none of them is repeated. The first labelled vertices form a clique and hence the rank of the adjacency matrix of the -vertex which is is at least .
The bound in Theorem 5.2 is reached, for instance, by the threshold graphs (the complete graph) and by .
Theorem 5.3. Let be a threshold graph on vertices, with Durfee square size and nullity . If , then has duplicate vertices.
Proof. The last rows of are not affected by the introduction of the zero diagonal when constructing as in Theorem 4.1. Hence, duplicates may only occur among the last labelled vertices. If were to have no duplicate vertices, then the last rows of need to be all different. Since the row is long, then, by a form of the pigeon-hole principle, the largest number of vertices possible for the graph to have no duplicates is . Therefore if , has at least one pair of duplicate vertices.
A threshold graph may have duplicate vertices even if . 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 of a threshold graph is the number of vertices removed to obtain a canonical graph.
Proof. Let be the canonical graph obtained from by removing all the duplicate vertices. Let us say that the number of vertices removed is . Since the reflection in the first column of the adjacency matrix of is in row echelon form, then the rows of after the is in strict “stepwise” form. Hence, the columns of are linearly independent. Now if the vertices are added to in turn to obtain 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 linear combinations among the rows of arising from the repeated rows in the last rows. Therefore, the nullity of is . 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 rows of give coduplicates. Also is the degree of a vertex in the first part of the equitable partition of the threshold graph defined by for Lab1. For , this corresponds to the degree in the monotonic nonincreasing sequence of distinct degrees (the 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 on an odd number of vertices has a duplicate vertex. (ii)An antiregular graph on an even number of vertices has a coduplicate vertex.
Proof. The graph is . Therefore if is even, it has a clique of two and hence a pair of coduplicate vertices. On the other hand, if 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 and , respectively, to obtain a canonical graph.
Theorem 5.7. A threshold graph with nondegenerate representation , where is even, has(i) duplicate vertices,(ii) coduplicate vertices.For odd , has(i) duplicate vertices,(ii) coduplicate vertices.
6. Minimal Configurations
Most of the information to determine the grounds for a labelled graph to be singular is encoded in the nullspace of its adjacency matrix (i.e., in ). The support of a kernel eigenvector in is the set of vertices corresponding to the nonzero entries. These vertices induce a subgraph termed the core of with respect to . Therefore a core of with respect to is a core graph in its own right. The size of the support is said to be the core order .
Definition 6.1 (see ). Let be a core graph on at least two vertices, with nullity and a kernel eigenvector having no zero entries. If a graph , of nullity one, having as the nonzero part of the kernel eigenvector, is obtained by adding independent vertices, whose neighbours are vertices of , then is said to be a minimal configuration (MC) with core .
Hence, an MC with core is a connected singular graph of nullity one having a minimal number of vertices and edges for the core , satisfying . The MCs may be considered as the “atoms” of a singular graph [19, 20]. The smallest MC is corresponding to a pair of duplicates. For core order three, the only MC is . The number of MCs increases fast for higher core order (see e.g., ). Figure 7 shows two graphs, (a) , the only MC with core and (b) a nut graph of order seven .
A basis for the nullspace of the adjacency matrix of a graph of nullity can take different forms. We choose a minimal basis for the nullspace of , that is, a basis having a minimal total number of nonzero entries in its vectors [19, 22].
Such a minimal basis for 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 , also ordered according to the nonincreasing core orders. Moreover, for all , the entry of the core order sequence for , does not exceed the entry of any other core order sequence of the graph. We say that the vectors in define a fundamental system of cores of , consisting of a collection of cores of minimal core order corresponding to a basis of linearly independent nullspace vectors . The significance of MCs can be gauged from the next result.
Theorem 6.2 (see [19, 20]). Let be a singular graph of nullity . There exist MCs which are subgraphs of whose core vertices are associated with the nonzero entries of the vectors in a minimal basis of the nullspace of .
From Theorem 5.4, the following result follows immediately.
Corollary 6.3. The only MC found in a threshold graph as a subgraph is .
Corollary 6.3 has been generalized to cographs in ; that is, in cographs, only (corresponding to duplicate vertices) may be found as an MC corresponding to a vector in . Therefore it is sufficient to have just 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 as an induced subgraph.
Proof. Suppose an MC is -free. Then, it is a cograph. Therefore, the only MC to contribute to the nullity is of core order two. We deduce that all other MCs, which have core order at least three, are not cographs.
Since is self-complementary, it follows that the complement of an MC with core order at least three also has as an induced subgraph. Figures 8(b) and 8(c) show as an induced subgraph (dotted edges) of the MC .
The second largest eigenvalue of is the golden section . By interlacing, we obtain the following result.
Theorem 6.5. The second largest eigenvalue of an is bounded below by .
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 are closely related to the number of walks in . The product of those factors of the minimum polynomial of , corresponding to the main eigenvalues only, has interesting properties.
Definition 7.1. The polynomial whose roots are the main eigenvalues of the adjacency matrix of a graph is termed the main characteristic polynomial.
For a proof of the following result, see , for instance.
Lemma 7.2 (see , rowmain). The main characteristic polynomial has integer coefficients , for all , .
7.1. The Main Eigenvalues of Antiregular Graphs
Recall that has exactly one pair of either duplicates or coduplicates.
Theorem 7.3. All eigenvalues of other than 0 or −1 are main.
Proof. Let Prop be all eigenvalues of , other than 0 or −1, are main. We prove Prop by induction on .(i)Prop(2) refers to whose only nonmain eigenvalue is −1. Prop(3) refers to whose only nonmain eigenvalue is 0.
This establishes the base cases.
(ii)Assume that Prop is true for all . Therefore for a nonmain eigenvalue other than 0 or −1, implies for .(iii)Consider and let be its adjacency matrix.For the case when is odd and is connected, let for an eigenvalue and . It follows that, for , and, for , . Similar equations are obtained for the case when is even.
The eigenvalue is nonmain if and only if , whence or or .
If (labelled 1) is the dominating vertex of , then, by Proposition 4.5, .
If , then restricted to is an eigenvector for the same eigenvalue . Therefore, by the induction hypothesis . Hence, or . The result follows by induction on .
7.2. The Main Eigenvalues of Threshold Graphs
By Theorem 7.3, all eigenvalues of that are not 0 or −1 are main. We show that this is still the case for a threshold graph having and for obtained from the degenerate form by adding duplicates and/or coduplicates.
Lemma 7.4. A graph has the same number of main eigenvalues as its complement.
Proof. Let be the adjacency matrix of the complement of a graph and the matrix with each entry equal to one. Then, . Now is a nonmain eigenvalue of if and only if . Hence, and share the same eigenvectors only for nonmain eigenvalues.
Theorem 7.5. Let be a threshold graph. All eigenvalues, other than 0 or −1, are main.
Proof. Let be for . Let the proposition Prop be all eigenvalues of other than 0 or −1, are main. We prove Prop by induction on .(i)If , then is not regular. Hence, the number of main eigenvalues is at least two. The other distinct eigenvalues, 0 and/or −1, are nonmain. By Theorem 5.7, has at least nonmain eigenvalues equal to 0 or −1. Thus, the number of main eigenvalues of is two. This establishes the base case, namely, Prop(2).(ii)The induction hypothesis is as follows: assume that Prop is true.(iii)We show that this is also true for a nondegenerate .The complement of is . By Lemma 7.4, and have the same number of main eigenvalues. One of the isolated vertices in contributes to the number of main eigenvalues. By the induction hypothesis, has main eigenvalues and nonmain eigenvalues. Hence, has main eigenvalues. The result follows by induction on .
We deduce immediately a spectral property of a threshold graph and its underlying antiregular graph.
Corollary 7.6. The nondegenerate threshold graph and its underlying have and main eigenvalues, respectively.
An equitable partition of the vertex set of a graph satisfies , where X is the indicator matrix whose column is the characteristic 0-1-vector associated with the part, containing entries equal to 1. The matrix turns out to be the adjacency matrix of the quotient graph (also known as divisor).
Lemma 7.7. The main part of the spectrum of is included in the spectrum of .
Proof. Let be a main eigenvalue of . Then, , where . Since , so that . Thus, the eigenvalue of is also an eigenvalue of , provided that . Indeed this is the case when is a main eigenvalue, since . Thus, the main part of the spectrum of is contained in the spectrum of .
We now show that the main part of the spectrum of is precisely the spectrum of . Consider the equitable vertex partition for as outlined in Section 3.2.
Theorem 7.8. Let the threshold graph have duplicates, coduplicates, and an equitable partition corresponding to the parts . Let be the adjacency matrix of the quotient graph . Then, , where is the main characteristic polynomial of .
Proof. The vertex labelling Lab1 is used. Let the vertices be labelled in order starting from those corresponding to , followed by those for and so on. If X is the indicator matrix whose column is the characteristic 0-1-vector associated with containing exactly nonzero entries (each equal to 1), then , where is . Now, by Theorem 7.5, in a threshold graph, 0 and −1 are the only nonmain eigenvalues and these correspond to duplicates and coduplicates, respectively. Therefore, the number of main eigenvalues of is exactly . Since the main spectrum of is contained in the spectrum of and is , then the roots of are the main eigenvalues of .
We give an example to clarify the procedure. Consider the threshold graph (Lab1), of Figure 3. We use the adjacency matrix and indicator matrix , indexed according to Lab2: The rows of are the distinct rows of . Therefore, Its spectrum is 7.16, 0.892, 0.448, −1.40, −1.59, −2.50, which is precisely the main part of the spectrum of .
For , the entries of give the number of walks of length from each vertex of . The matrix whose column is is denoted by . The dimension of the subspace generated by the columns of is the rank of .
Theorem 7.9 (see ). For a graph with main eigenvalues, the rank, , of the matrix is , for .
The columns are a maximal set of linearly independent vectors in . Thus, the first columns provide all the information on the number of walks from each vertex of any length .
Definition 7.10. The matrix of rank is said to be the walk matrix .
Note that has the least number of columns for a walk matrix to reach the maximum rank possible which is . From Corollary 7.6, has main eigenvalues.
Theorem 7.11. The rank of the walk matrix of is .
The number of walks of length can be expressed in terms of the main eigenvalues [28, page 46].
Theorem 7.12. The number of walks of length starting from any vertex of is given by where is independent of for each and are the main eigenvalues of .
Since 0 is never a main eigenvalue of , it follows that all the main eigenvalues of contribute to the number of walks.
7.3. Cases of Reducible Main Polynomial
By Theorem 7.3, only one eigenvalue of is not main. Recall that the minimal equitable vertex partition of satisfies , where is the adjacency matrix of the quotient graph and , the main characteristic polynomial of .
We note that for many threshold graphs is irreducible over the integers. For example the only eigenvalue of (in degenerate form) which is not main is −1 and , which is irreducible.
Now we add vertices to the degenerate form . If we add a vertex to the first part, to obtain , a negative eigenvalue (not −1) and 0 appear. The eigenvalue −1 is lost and . When a vertex is added to the third part to obtain , the eigenvalue −1 is retained while the zero eigenvalue appears and . In both these latter two cases is irreducible over the integers. Now when a vertex is added to the seventh part to obtain , the eigenvalue −1 is retained while the zero eigenvalue appears. In this case, however, , and therefore it is reducible over the integers.
This is also the case for some instances of the threshold graphs when the cubic polynomial has an integer as a root and therefore is reducible. The divisor is with characteristic polynomial .
If is 0, 2 or 3, there are no integral values of and satisfying the polynomial . If , the graph either for and or for and satisfies it. Also for either the graph for and , or and , or and satisfies it, while for , the graph for and 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  it was remarked that an antiregular graph has a bipartite character, that is, the number of negative eigenvalues is equal to the number of positive ones. We denote the number of zero eigenvalues by .
8.1. The Spectrum of
For , is not bipartite. Therefore, . 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 to (according to the labelling (Lab2) of the stepwise adjacency matrix),(i)a vertex duplicate to the is added for even ,(ii)a vertex coduplicate to the is added for odd .
Theorem 8.2. for .
Proof. The proof is by induction on .
The spectra of the three smallest antiregular graphs, , , and , establish the base cases.
Assume that the theorem is true for .
We prove it true for .
If is singular, then it has a duplicate vertex and is odd. By the induction hypothesis .
If, on the other hand, is nonsingular, then has a coduplicate vertex and is even. Again the nonzero eigenvalues satisfy .
We apply Lemma 8.1, using Lab2. For odd , if a vertex , coduplicate to the vertex, is added to , then only one of the duplicate vertices of will have as a neighbour in . The zero eigenvalue of vanishes and the eigenvalue −1 is introduced for . By the Perron Frobenius theorem adding edges to a graph () increases the maximum eigenvalue. Therefore, by interlacing, the number of positive eigenvalues increases by one. Since the new coduplicate vertex contributes the new eigenvalue −1 to the spectrum, it follows that will be satisfied in . 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 , if a vertex , duplicate to the vertex, is added, then a duplicate vertex is added to the graph, retaining .
The result follows by induction on .
8.2. The Spectrum of a Threshold Graph
In this section, we shall represent the antiregular graph by the degenerate form . As in Section 4, any part can be expanded to produce a threshold graph . 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 of the adjacency matrix increases, then, by interlacing, the number of eigenvalues less than and the number greater than remain the same; (ii) the multiplicity of an eigenvalue of the adjacency matrix decreases, then by interlacing, each of the numbers and increases by one.
We shall write for and for .
First we see an application of Lemma 8.3(i) using Lab1. For even , if one of the even indexed , for , of is increased, then a coduplicate of a vertex is added. This forces and to remain unchanged while each of and the multiplicity of the eigenvalue −1 increases by one. If the odd indexed , for some , is increased, then a duplicate of a vertex is added forcing and to remain unchanged.
Similarly, for odd , if the even indexed , for some , is increased, then a duplicate of a vertex is added. This forces and to remain unchanged while increases by one. If the odd indexed , for some , is increased, then a coduplicate of a vertex is added forcing and to remain unchanged.
The case for even and is the same as for odd with (Lab1). Taking for odd with and expanding to with gives the unique case where decreases by one and increases by one. Since decreases by one, by Lemma 8.3(ii), each of and increases by one, the latter corresponding to the increase in the multiplicity of the eigenvalue −1. We have proved the following result.
Theorem 8.4. If the threshold graph is transformed to another threshold graph by increasing exactly one of the s by one, then
The simple graphic appeal of the Ferrers/Young diagram , with rows representing the degree sequence of a -vertex threshold graph has been instrumental to obtain interesting results on the nullity and structure of the graph. The shape of has been also used to determine the nature of the eigenvalues as main or nonmain.
Let be the diagonal entries whose nonzero entries are the vertex degrees for some labelling of the vertices. Like the adjacency matrix , the Laplacian 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 (i.e., the lengths of the columns of ) 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 .
This paper was supported by the Research Project Funds MATRP01-01 Graph Spectra and Fullerene Molecular Structures of the University of Malta.
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