Abstract
The Kirchhoff index of is the sum of resistance distances between all pairs of vertices of in electrical networks. is the Laplacian-Energy-Like Invariant of in chemistry. In this paper, we define two classes of join graphs: the subdivision-vertex-vertex join and the subdivision-edge-edge join . We determine the generalized characteristic polynomial of them. We deduce the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomials of and when is -regular graph and is -regular graph. As applications, the Laplacian spectra enable us to get the formulas of the number of spanning trees, Kirchhoff index, and of and in terms of the Laplacian spectra of and .
1. Introduction
Let be a simple graph on vertices and edges. Let be the degree of vertex in and be the diagonal matrix with diagonal entries . Let denote the adjacency matrix of a graph . The Laplacian matrix and the signless Laplacian matrix of are defined as and , respectively. Let , or simply ( and , resp.), be the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomial of and its roots be the adjacency (Laplacian and signless Laplacian, resp.) eigenvalues of , denoted by ( and , resp.). The line graph of is denoted by .
The generalized characteristic polynomial of , introduced by Cvetkovi et al. [1], is defined to be . The generalized characteristic polynomial covers the cases of usual characteristic polynomial and Laplacian and signless Laplacian polynomials of graph , due to variation of the parameter and, whenever necessary, replacing by . We can get that the characteristic polynomials of , , and are equal to , , and .
Let be a connected graph. For two vertices and of , the resistance distance between and is defined to be the effective resistance between them when unit resistors are placed on every edge of . It is a distance function on graphs introduced by Klein and Randić [2]. The Kirchhoff index of , denoted by , is the sum of resistance distances between all pairs of vertices of . For a connected graph of order [3], . Recently, many results on Kirchhoff index were obtained [2, 4–8]. Laplacian-Energy-Like Invariant was named in [9]. The motivation for introducing LEL was in its analogy to the earlier studied graph energy and Laplacian energy [10]. Although Kirchhoff index and LEL both depend on Laplacian eigenvalues, their comparative study started only quite recently [11, 12].
Graph operations, such as the disjoint union, the join, the corona, the edge corona, and the neighborhood corona [13–17], are techniques to construct new classes of graphs from old ones. In [18], a real molecular graph of ferrocene is a join graph obtained from graphs and , where is a disjoint union of two pentagons. In [4–6, 8, 14] the resistance distance and Kirchhoff index of artificial graphs are computed. Although most of the constructed graphs in the literature are contrived, they may be of use for chemical and physical applications.
Motivated by the work above, we define two new graph operations based on subdivision graphs as follows.
The subdivision graph of a graph is the graph obtained by inserting a new vertex into every edge of [19]. We denote the set of such new vertices by . In [15, 20], some new graph operations based on subdivision graphs were defined and the -, -, and -spectrum were computed in terms of those of the two graphs.
Let and be two vertex disjoint graphs. The subdivision-vertex-vertex join of and , denoted by , is the graph obtained from and by joining every vertex in to every vertex in . The subdivision-edge-edge join of and , denoted by , is the graph obtained from and by joining every vertex of to every vertex in .
Note that if is a graph on vertices and edges and is a graph on vertices and edges, then has vertices and edges and has vertices and edges.
Let denote a cycle of order and denote a path of order . Figure 1 depicts the subdivision-vertex-vertex join and subdivision-edge-edge join , respectively.
The paper is organized as follows. In Section 2, some useful lemmas are provided. In Section 3, we compute the generalized characteristic polynomial of the subdivision-vertex-vertex join and obtain the -, -, and -spectrum in terms of the corresponding spectrum of and when and are regular graphs. In Section 4, we compute the generalized characteristic polynomial of the subdivision-edge-edge join and obtain the -, -, and -spectrum in terms of the corresponding spectrum of and when and are regular graphs. In Section 5, as the applications, the number of spanning trees, Kirchhoff index, and LEL of the subdivision-vertex-vertex join and the subdivision-edge-edge join graphs are computed. In Section 6, we conclude the paper.
2. Preliminaries
The -coronal of an matrix is defined [16] to be the sum of entries of the matrix ; that is where denotes the column vector of size when all the entries are equal to one.
Lemma 1 (see [21]). Let be an real matrix and let denote the matrix with all entries equal to one. Then where is a real number and is the adjugate matrix of . Moreover,
Lemma 2 (see [13]). If is an matrix when each row sum is equal to a constant ; then,
Lemma 3 (see [22]). Let , , , and be, respectively, , , , and matrices with and invertible. Then where and are called the Schur complements of and , respectively.
Lemma 4 (see [19]). If is a regular graph of degree , with vertices and edges; then
If is an matrix and is an matrix, then the Kronecker product is defined as the matrix with the block form
This is an associative operation with the property that and whenever the products and exist. The latter implies for nonsingular matrices and . Moreover, if and are and matrices, then . The reader is referred to [23] for other properties of the Kronecker product not mentioned here.
3. Spectrum of Subdivision-Vertex-Vertex Join
Let be an -regular graph on vertices and edges and an -regular graph on vertices and edges, respectively. We first label the vertices of as follows. Let , , , and . The adjacency matrix of can be written as follows: where denotes the matrix with all entries equal to zero, is the incidence matrix of , is the identity matrix of order , and is the column vector with all entries equal to 1. It is clear that the degrees of the vertices of are for , for , for , and for . Then the degree matrix of subdivision-vertex-vertex join can be written as follows:
3.1. Generalized Characteristic Polynomial of
Theorem 5. Let be an -regular graph on vertices and edges and be an -regular graph on vertices and edges. Then the generalized characteristic polynomial of subdivision-vertex-vertex join of and is
Proof. Let be the incidence matrix of . Then, with respect to the adjacent matrix and degree matrix of , the generalized matrix of is given byThus the generalized characteristic polynomial of iswhereis the Schur complement of . So, , whereis the Schur complement of . So, is the Schur complement of . So,by Lemma 1,
Theorem 6. Let be an -regular graph on vertices and edges and an -regular graph on vertices and edges. Then(1)the adjacency characteristic polynomial of is (2)the Laplacian characteristic polynomial of is (3)the signless Laplacian characteristic polynomial of is
4. Spectrum of Subdivision-Edge-Edge Join
The adjacency matrix of can be written as follows: where denotes the matrix with all entries equal to zero, is the incidence matrix of , is the identity matrix of order , and is the column vector with all entries equal to 1. It is clear that the degrees of the vertices of are for , for , for , and for . Then the degree matrix of subdivision-edge-edge join can be written as follows:
4.1. Generalized Characteristic Polynomial of
Theorem 7. Let be an -regular graph on vertices and edges and be an -regular graph on vertices and edges. Then the generalized characteristic polynomial of subdivision-edge-edge join of and is
Proof. Let be the incidence matrix of . Then, with respect to the adjacent matrix and degree matrix of , the generalized matrix of is given by Thus the generalized characteristic polynomial of is whereis the Schur complement of . So, , where is the Schur complement of . So, is the Schur complement of . So,by Lemma 4,and by Lemma 1,
Theorem 8. Let be an -regular graph on vertices and edges and an -regular graph on vertices and edges. Then(1)the adjacency characteristic polynomial of is (2)the Laplacian characteristic polynomial of is (3)the signless Laplacian characteristic polynomial of is
5. Applications
We give the number of spanning trees, the Kirchhoff index, and of the two classes of join graphs, as the applications.
We can easily compute the -spectrum of in terms of -spectrum of and by (2) of Theorem 6.
Corollary 9. Let be an -regular graph on vertices and edges and an -regular graph on vertices and edges. Then the Laplacian spectrum of consists of(1)2, repeated times;(2)two roots of the equation , for each ;(3)two roots of the equation , for each ;(4)four roots of the equation + + , and the four roots are , , , and , respectively.
We can easily compute the L-spectrum of in terms of -spectrum of and by (2) of Theorem 8.
Corollary 10. Let be an -regular graph on vertices and edges and an -regular graph on vertices and edges. Then the Laplacian spectrum of consists of(1), repeated times;(2), repeated times;(3)two roots of the equation , for each ;(4)two roots of the equation , for each ;(5)four roots of the equation + + , and the four roots are , , , and , respectively.
Let denote the number of spanning trees [1] of . It is well known that if is a connected graph on vertices with Laplacian spectrum , then .
Corollary 11. Let be an -regular graph on vertices and edges and an -regular graph on vertices and edges. The number of spanning trees of and are, respectively,
Proof. For , the number of the vertices is . By Corollary 9 and Vieta’s formulas, we can obtain the following: for every , ; for every , ; for the equation + , Therefore, we complete the proof of and we omit the proof of
The Kirchhoff index [24] is an important physical and chemical indicator. Let be a connected graph of order ; then
Corollary 12. Let be an -regular graph on vertices and edges and an -regular graph on vertices and edges; then the Kirchhoff index of and is, respectively,
Proof. For , by Corollary 9 and Vieta’s formulas, we can obtain the following: for every , ; for every , ; for the equation + + , Therefore, we complete the proof of and we omit the proof of
denotes Laplacian-Energy-Like Invariant [24]. Let be a connected graph of order with edges; then
Corollary 13. Let be an -regular graph on vertices and edges and an -regular graph on vertices and edges; then where , , and , , and are three roots of the equation + + . where , , and , , and are three roots of the equation + + − .
6. Conclusions
In this paper, two classes of join graphs, and , are constructed by graph operations. The formulas of generalized characteristic polynomial of the two graphs are obtained by using block matrix and Schur complement. For regular graphs and , we characterize the -, -, and -spectrum of and in terms of the corresponding spectrum of and , and by the -spectrum, we get the number of spanning trees, Kirchhoff index, and LEL. But for nonregular graphs and , we cannot determine the spectrum of and . The most difficult problem is that we cannot find the rule of the block matrices to deduce a simple result. Therefore, we should try to find new method to solve the problem.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This paper is supported by the National Natural Science Foundation of China (no. 11361033).