In this paper, a simple and efficient method is developed for evaluating the eigenvalues of matrices having special patterns. This is achieved by decomposing the matrices into specific forms. The application is extended to calculate the eigenvalues of the Laplacian of graphs having special connectivity properties.

The undirected power graph of a  nite group G, P(G), is a graph with the group elements of G as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let A be an adjacency matrix of P(G). An eigenvalue  of A is a main eigenvalue if the eigenspace "( ) has an eigenvector X such that Xtj 6= 0, where j is the all-one vector. In this paper we want to focus on the power graph of the  nite cyclic group Zn and  nd a condition on n where P(Zn) has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of P(Zn) where n has a unique prime decomposition n = prp2. We also formulate a conjecture on the number of the main eigenvalues of P(Zn) for an arbitrary positive integer n.

For a commutative ring R with identity 1 6= 0, let the set Z(R) denote the set of zero-divisors and let Z*(R) = Z(R) \{0} be the set of non-zero zero divisors of R. The zero divisor graph of R, denoted by Г,(R), is a simple graph whose vertex set is Z* (R) and two vertices u,v ,Z*(R) are adjacent if and only if uv = vu = 0. In this article, we _nd the signless Laplacian spectrum of the zero divisor graphs Г, (Zn) for n = pM1 qM2, where p < q are primes and M1, M2 are positive integers.

Set X = fM11; M12; M22; M23; M24; Zn; T4n; SD8n; Sz(q); G2(q); V8ng, where M11, M12, M22, M23, M24 are Mathieu groups and Zn, T4n, SD8n, Sz(q), G2(q) and V8n denote the cyclic, dicyclic, semi-dihedral, Suzuki, Ree and a group of order 8n presented by V8n = ha; b j a2n = b4 = e; aba = b􀀀 1; ab􀀀 1a = bi; respectively. In this paper, we compute all eigenvalues of Cay(G; T), where G 2 X and T is minimal, second minimal, maximal or second maximal normal subset of G n feg with respect to its size. In the case that S is a minimal normal subset of G n feg, the summation of the absolute value of eigenvalues, energy of the Cayley graph, is evaluated.

Let $G$ be a simple graph with order $n$ and size $m$. Let $D(G)=$ diag$(d_1, d_2, \dots, d_n)$ be its diagonal matrix, where $d_i=\deg(v_i),$ for all $i=1,2,\dots,n$ and $A(G)$ be its adjacency matrix. The matrix $Q(G)=D(G)+A(G)$ is called the signless Laplacian matrix of $G$. Let $q_1,q_2,\dots,q_n$ be the signless Laplacian eigenvalues of $Q(G)$ and let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of the $k$ largest signless Laplacian eigenvalues. Ashraf et al. [F. Ashraf, G. R. Omidi, B. Tayfeh-Rezaie, On the sum of signless Laplacian eigenvalues of a graph, Linear Algebra Appl. {\bf 438} (2013) 4539-4546.] conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $k=1,2,\dots,n$. We present a survey about the developments of this conjecture.

A useful tool for investigation various problems in mathematical chemistry and  computational physics is graph entropy.  In this paper, we introduce a new version of graph entropy  and  then we determine it for some classes of graphs.

In this paper, a fundamentally new method, based on the definition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. Some examples are provided to show the accuracy and reliability of the proposed method. It is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to the desired eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. These examples show an interesting phenomenon in the procedure: The diagonal matrix that converges to eigenvalues gives them in decreasing order in the sense of absolute value. Appendices A to C provide Matlab codes that implement the proposed algorithms. They show that the proposed algorithms are very easy to program.

So far, various components of image characteristics have been used for steganal-ysis, including the histogram characteristic function, adjacent colors distribution, and sample pair analysis. However, some certain steganography methods have been proposed that can thwart some analysis approaches through man- aging the embedding patterns. In this regard, the present paper is intended to introduce a new analytical method for detecting stego images, which is robust against some of the embedding patterns designed specifically to foil steganalysis attempts. The proposed approach is based on the analysis of the eigenvalues of the cover correlation matrix used for the purpose of the study. Image cloud partitioning, vertical correlation function computation, constellation of the correlated data, and eigenvalues examination are the major challenging stages of this analysis method. The proposed method uses the LSB plane of images in spatial domain, extendable to transform domain, to detect low embedding rates-a major concern in the area of the LSB steganography. The simulation results based on deviation detection and rate estimation methods indicated that the proposed approach outperforms some well-known LSB steganalysis methods, specically at low embedding rates.

In this paper, we determine the distance Laplacian and distance signless Laplacian spectrum of generalized wheel graphs and a new class of graphs called dumbbell graphs.

The aim of this paper is to determine an upper bound for the number of non-co-spectral permutation graphs in terms of automorphism group of a graph G. As a corollary, we determine the eigenvalues of all permutation graphs P (Cn), where  2 Aut(Cn).