On the Adjacency, Laplacian, and Signless Laplacian Spectrum of Coalescence of Complete Graphs

Abstract

Coalescence as one of the operations on a pair of graphs is significant due to its simple form of chromatic polynomial. The adjacency matrix, Laplacian matrix, and signless Laplacian matrix are common matrices usually considered for discussion under spectral graph theory. In this paper, we compute adjacency, Laplacian, and signless Laplacian energy ( energy) of coalescence of pair of complete graphs. Also, as an application, we obtain the adjacency energy of subdivision graph and line graph of coalescence from its energy.

1. Introduction

Throughout the discussion by a graph we mean simple graph without self loops or multiple edges. Let be a simple graph on vertices with vertex set. The line graph of a graph is the graph with vertex set as edge set of with two vertices (edges of G) adjacent if and only if they are having a vertex in common. Similarly, the subdivision graph of a graph is the graph obtained by inserting a vertex of degree two in each edge of . The adjacency matrix of denoted by is a matrix, where if vertex is adjacent to and 0 otherwise. Clearly, is real symmetric so that eigenvalues of which are roots of its characteristic equation given by are real. They are denoted by and can be arranged in descending order as . The spectrum of is collection of eigenvalues along with their multiplicity and energy of a graph is simply defined as . For more details and rigorous treatment on adjacency spectra and energy, see [1–4]. Let denote the degree of a vertex which is the number of edges incident on it. The degree matrix is a diagonal matrix having diagonal entry as the degree of the corresponding vertex. We denote the average degree of a graph , as The matrix is called Laplacian matrix. The roots of the characteristic polynomial of the Laplacian matrix are called Laplacian eigenvalues denoted by . The matrix is also real symmetric but singular so eigenvalues can be arranged as . The Laplacian spectrum of is the collection of Laplacian eigenvalues along with their multiplicity and Laplacian energy is defined as .

For an extensive literature on Laplacian spectra and energy, one can refer to [5–11]. On similar lines, the signless Laplacian matrix of a graph is defined as . The signless Laplacian eigenvalues are also real and can be denoted by. The signless Laplacian energy (or simply energy) is defined similar to Laplacian energy as .

Some results on signless Laplacian energy are available in [12–15]. The Laplacian and signless Laplacian eigenvalues for a connected graph are nonnegative.

Let and be graphs on disjoint sets of vertices, respectively. Suppose is a clique in and is a clique in . Let be a graph obtained from and by identifying (coalescing into a single vertex) and . Then, is an overlap of and in . It may be viewed as generalized coalescence denoted by .

The structure of depends on vertices chosen for overlap. Its chromatic polynomial can be split into chromatic polynomials of and (see [16]). If , we call it vertex coalescence denoted by and for we call it edge coalescence denoted by . For vertex coalescence of two graphs and , the adjacency matrix has the form

Similarly, the edge coalescence of two graphs and has the adjacency matrix structurewhere is matrix of all 1’s and is the matrix of all zeros having appropriate order.

2. Results

2.1. Adjacency Energy

Theorem 1. The energy (adjacency energy) of is given by , where , , and are the roots of the cubic:

Proof. The coalescence of the complete graphs and at a point results in a graph with vertices and edges. The adjacency matrix takes the form so that characteristic polynomial isBy performing and in succession, we haveOn simplifying, finally we getFrom this equation, the theorem follows.

Theorem 2. The energy (adjacency energy) of edge coalescence of complete graphs is given by , where , , and are the roots of the cubic equation:

Proof. The coalescence of the complete graphs and on an edge results in a graph with vertices and edges. The adjacency matrix takes the formso that the characteristic polynomial of the edge coalescence isBy performing for we haveBy performing for we haveOn expanding and simplifying, we get the required polynomial and hence the theorem.

2.2. Laplacian Energy

Now we discuss the Laplacian energy of coalescence.

Lemma 3 (see [17]). If G is any connected graph of order n with Laplacian eigenvalues with , then, the number of spanning trees of G is given by

Theorem 4. The Laplacian energy of the vertex coalescence of complete graphs and is given by

Proof. The degree matrix of the vertex coalescence with suitable labeling has the formThe adjacency matrix isThe Laplacian matrix now becomesso that the Laplacian polynomial isPerformingwe get Again performingand directly expanding along first column, we getSo that the Laplacian spectrum is times, times, , , and .
Now the avdhence, the Laplacian energy becomes

Corollary 5. The number of spanning trees of according to Lemma 3 is as expected since has a cut point with number of spanning trees and in each block (complete graph) separately.

Theorem 6. The Laplacian energy of the edge coalescence of complete graphs and is given bywhere , , and are the roots of the cubic equation:

Proof. The degree matrix of the edge coalescence with suitable labeling has the formThe adjacency matrix isThe Laplacian polynomial is then given byPerforming,Again performing and directly expanding giveOn simplifying, we get the Laplacian polynomial asOn equating to zero and extracting eigenvalues from the equation above, the theorem follows.
Note. When , the Laplacian polynomial isThe Laplacian eigenvalues are , times, twice, 2, and 0 once.
Sincethe Laplacian energy is

2.3. Signless Laplacian Energy

Now we consider the case of signless Laplacian matrix of the coalescence of complete graphs and deduce the corresponding energy. Before we do so, consider the following results.

Lemma 7 (see [17]). If G is any graph with p vertices and q edges, then characteristic polynomial of line graph in terms of (signless Laplacian) polynomial is given by

Lemma 8 (see [18]). If G is any graph with p vertices and q edges, then characteristic polynomial of subdivision graph in terms of (signless Laplacian) polynomial is given by

Theorem 9. The signless Laplacian energy of the vertex coalescence of complete graphs and is given bywhere

Proof. From the degree matrix and adjacency matrix of the vertex coalescence , we have the signless Laplacian matrix:The signless Laplacian polynomial is thenPerforming,Again performingthen directly expanding along first column, we obtainThe signless Laplacian eigenvalues are times, times,, and
Now thehence, the theorem follows.

Corollary 10. From Lemma 7, the energy (adjacency energy) of line graph of is given by In particular for ,

Corollary 11. From Lemma 8, the energy (adjacency) of subdivision graph of where , is given by

Theorem 12. The signless Laplacian energy of the edge coalescence of complete graphs and is given bywhere , and are roots of the cubic equation:

Proof. From the degree and adjacency matrix, the signless Laplacian matrix of the edge coalescence isThe signless Laplacian polynomial is then given byPerforming,Again performingand expanding directly yieldOn performing elementary operations, we finally arrive atOn equating to zero and extracting eigenvalues from the equation above, the theorem follows.

Corollary 13. From Lemma 7, the energy (adjacency) of line graph of is given bywhere, , , and are the roots of the cubic equation:

Corollary 14. From Lemma 8, the energy (adjacency) of subdivision graph of where is given by where , , and are the roots of the equation

Competing Interests

The authors declare that they have no competing interests.