Independent sets of some graphs associated to commutative rings

Let $G=(V,E)$ be a simple graph. A set $S\subseteq V$ is independent set of $G$, if no two vertices of $S$ are adjacent. The independence number $\alpha(G)$ is the size of a maximum independent set in the graph. %An independent set with cardinality Let $R$ be a commutative ring with nonzero identity and $I$ an ideal of $R$. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is an undirected graph whose vertices are the nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. Also the ideal-based zero-divisor graph of $R$, denoted by $\Gamma_I(R)$, is the graph which vertices are the set {x\in R\backslash I | xy\in I \quad for some \quad y\in R\backslash I\} and two distinct vertices $x$ and $y$ are adjacent if and only if $xy \in I$. In this paper we study the independent sets and the independence number of $\Gamma(R)$ and $\Gamma_I(R)$.Comment: 27 pages. 22 figure

Centrosymmetric graphs and a lower bound for graph energy of fullerenes

The energy of a molecular graph G is defined as the summation of the absolute values of the eigenvalues of adjacency matrix of a graph G. In this paper, an infinite class of fullerene graphs with 10n vertices, n ≥ 2, is considered. By proving centrosymmetricity of the adjacency matrix of these fullerene graphs, a lower bound for its energy is given. Our method is general and can be extended to other class of fullerene graphs

Balaban Index of an Infinite Class of Dendrimers

The Balaban index of a graph G is the first simple index of very low degeneracy. It is defined as the sum of topological distances from a given atom to any other atoms in a molecule. In this paper the Balaban index of an infinite family of dendrimers is computed. The result can be of interest in molecular data mining, particularly in searching the uniqueness of tested (hyper-branched) molecular graphs

Automorphism Group of Certain Power Graphs of Finite Groups

The power graph $\mathcal{P}(G)$ of a group $G$ is the graphwith group elements as vertex set and two elements areadjacent if one is a power of the other. The aim of this paper is to compute the automorphism group of the power graph of several well-known and important classes of finite groups

ON SPECTRUM OF I-GRAPHS AND ITS ORDERING WITH RESPECT TO SPECTRAL MOMENTS

Suppose $G$ is a graph, $A(G)$ its adjacency matrix, and $μ_1(G), μ_2(G), \cdots, μ_n(G)$ are eigenvalues of $A(G)$. The numbers $S_k(G) = \sum_{i=1}^n μ^k_i(G)$, $0 \leq k \leq n − 1$, are said to be the k−th spectral moment of $G$ and the sequenceS(G) = (S_0(G), S_1(G), \sdots, S_{n−1}(G)) is called the spectral moments sequence of $G$. For two graphs $G_1$ and $G_2$, we define $G_1 \leq_S G_2$, if there exists an integer$k$, $1 \leq k \leq n − 1$, such that for each $i$, $0 \leq i \leq k − 1$, $S_i(G_1) = S_i(G_2)$ andS_k(G_1) < S_k(G_2).The I−graph $I(n, j, k)$ is a graph of order $2n$ with the vertex and edge sets$V(I(n, j, k) = \{u_0, u_1, \cdots, u_{n−1}, v_0, v_1, \cdots, v_{n−1}\}$,$E(I(n, j, k) = \{u_iu{u+j}, u_iv_i, v_iv_{i+k} ; 0 \leq i \leq n − 1\}$,respectively. The aim of this paper is to compute the spectrum of an arbitraryI−graph and the extremal I−graphs with respect to the S−order