An expansion graph of a directed weighted graph G0 is obtained fromG0 by replacing some edges by disjoint chains. The adjacency matrix of an expansion graph is a partial linearization of a matrix polynomial with nonnegative coefficients. The spectral radii for different expansion graphs of G0 and correspondingly the spectral radii of matrix polynomials with nonnegative coefficients, which sum u...

An expansion graph of a directed weighted graph G0 is obtained fromG0 by replacing some edges by disjoint chains. The adjacency matrix of an expansion graph is a partial linearization of a matrix polynomial with nonnegative coefficients. The spectral radii for different expansion graphs of G0 and correspondingly the spectral radii of matrix polynomials with nonnegative coefficients, which sum u...

Objective: The symmetric encryption technique is one of the most important fields for securing communications between people. In order to produce complex ciphertext, we are introducing new enciphering with help Hamiltonian path, a self-invertible key matrix and decryption. Methods: There many kinds methods, like Caesar Cipher, Atbash Hill etc. All these methods use common encryption, while decr...

While powers of the adjacency matrix of a finite graph reveal information about walks on the graph, they fail to distinguish closed walks from cycles. Using elements of an appropriate commutative, nilpotentgenerated algebra, a “new” adjacency matrix can be associated with a random graph on n vertices and |E| edges of nonzero probability. Letting Xk denote the number of k-cycles occurring in a r...

We give two proofs that, for a nite regular graph, the reciprocal of Ihara's zeta function can be expressed as a simple polynomial times a determinant involving the adjacency matrix of the graph. The rst proof is based on representing radial symmetric eigenfunctions on regular trees in terms of certain polynomials. The second proof is a consequence of the fact that the resolvent of the adjacenc...

let $d$ be a digraph with skew-adjacency matrix $s(d)$‎. ‎the skew‎ ‎energy of $d$ is defined as the sum of the norms of all‎ ‎eigenvalues of $s(d)$‎. ‎two digraphs are said to be skew‎ ‎equienergetic if their skew energies are equal‎. ‎we establish an‎ ‎expression for the characteristic polynomial of the skew‎ ‎adjacency matrix of the join of two digraphs‎, ‎and for the‎ ‎respective skew energ...

Graph properties suitable for the classification of instance hardness for the NP-hard MAX-CUT problem are investigated. Eigenvalues of the adjacency matrix of a graph are demonstrated to be promising indicators of its hardness as a MAX-CUT instance.

Let G^s be a signed graph, where G = (V;E) is the underlying simple graph and s : E(G) to {+, -} is the sign function on E(G). In this paper, we obtain k-th signed spectral moment and k-th signed Laplacian spectral moment of Gs together with coefficients of their signed characteristic polynomial and signed Laplacian characteristic polynomial are calculated.

Given a graph G; let kGk denote the trace norm of its adjacency matrix, also known as the energy of G: The main result of this paper states that if G is a graph of order n; then kGk + G (n 1) 1 +n ; where G is the complement of G: Equality is possible if and only if G is a strongly regular graph with parameters (n; (n 1) =2; (n 5) =4; (n 1) =4) ; known also as a conference graph. In fact, the a...

Graph isomorphism problem is to determine whether two given graphs are isomorphic. It is a particular type of a more general problem “the isomorphism of incidence system”. I propose some new invariants for heuristic search of graph isomorphism and demonstrate that they are really useful in practice by experiments. Keyword. graph isomorphism, graph invariant, graph spectrum , adjacency matrix, m...