Spectral Invariants and Their Application on Spectral Characterization of Graphs

Abstract

In this paper, we give a method to characterize graphs determined by their adjacency spectrum. At first, we give two parameters Π1(G) and Π2(G), which are related to coefficients of the characteristic polynomial of graph G. All connected graphs with Π1(G)∈{1,0,−1,−2,−3} and Π2(G)∈{0,−1,−2,−3} are characterized. Some interesting properties of Π1(G) and Π2(G) are also given. We then find the necessary and sufficient conditions for two classes of graphs to be determined by their adjacency spectrum.

Keywords

• characteristic polynomial
• cospectral graph
• invariant