Characterization of graphs whose signature equals the number of odd cycles

Let G be a simple graph with vertex set V(G) and edge set E(G). The signature s(G) of G is the difference between the number of positive eigenvalues and the number of negative eigenvalues of the adjacency matrix A(G). In [20], it was proved that −c1(G)≤s(G)≤c1(G), where c1(G) denotes the number of odd cycles in G. A problem arises naturally: What graphs have signature attaining the upper bound c1(G) (resp., the lower bound −c1(G))? In this paper, we focus our attention on this problem, characterizing graphs G whose signature equals c1(G) (resp., −c1(G)).