A new method for the sum-edge characteristic polynomials of graphs

Abstract

In this paper, the determinant of the sum-edge adjacency matrix of any given graph without loops is calculated by means of an algebraic method using spanning elementary subgraphs and also the coefficients of the corresponding sum-edge characteristic polynomial are determined by means of the elementary subgraphs. Also we gave a formula for the number of smallest odd-sized cycles in a given regular graph.