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

Mert Sinan Oz,Ismail Naci Cangul

doi:10.5269/bspm.47918

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.

Highlights

We will call the characteristic polynomial of S(G) as the sum-edge characteristic polynomial of G and denote it by PGse(x)
If the number of vertices of an elementary subgraph is equal to the number of vertices of G, the elementary subgraph is called a spanning elementary subgraph. c−(G′t) and c◦(G′t) are defined as the number of components in a subgraph G′t which are edges and cycles, respectively
Note that in a cycle component of the spanning elementary subgraph G′t of G, if vi and vj are two adjacent vertices in this cycle component, when calculating the latter product in Theorem 3.2 corresponding to cycle component, we consider only one of the edges vivj and vjvi due to the permutation structure

Summary

Introduction

We will call the characteristic polynomial of S(G) as the sum-edge characteristic polynomial of G and denote it by PGse(x). C−(G′t) and c◦(G′t) are defined as the number of components in a subgraph G′t which are edges and cycles, respectively. We can express the formula as follows: Let S(G) = [sij]n×n be the sum-edge adjacency matrix of G. Let ui and uj be two adjacent vertices in an edge component of corresponding elementary (spanning elementary) subgraph G′t of G.

Results

Conclusion