Discrete Dynamics in Nature and Society

Volume 2017, Article ID 2372931, 10 pages

https://doi.org/10.1155/2017/2372931

## Generalized Characteristic Polynomials of Join Graphs and Their Applications

School of Computer and Communication, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Correspondence should be addressed to Pengli Lu; moc.361@88ilgnepul

Received 11 November 2016; Revised 18 January 2017; Accepted 26 January 2017; Published 2 March 2017

Academic Editor: Francisco R. Villatoro

Copyright © 2017 Pengli Lu et al. 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 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.