Bounds for the positive and negative inertia index of a graph

Abstract

Let G be a graph and let A(G) be adjacency matrix of G. The positive inertia index (respectively, the negative inertia index) of G, denoted by p(G) (respectively, n(G)), is defined to be the number of positive eigenvalues (respectively, negative eigenvalues) of A(G). In this paper, we present the bounds for p(G) and n(G) as follows:m(G)−c(G)≤p(G)≤m(G)+c(G),m(G)−c(G)≤n(G)≤m(G)+c(G), where m(G) and c(G) are respectively the matching number and the cyclomatic number of G. Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds respectively.