On graphs with girth g and positive inertia index of [g]/2 1 and [g]/2

Let G be a graph. The numbers of positive, negative and zero eigenvalues (including multiplicities) of the adjacency matrix A(G) are denoted by p(G), n(G) and eta(G), respectively. Traditionally, p(G) (resp. n(G)) is called the positive (resp. negative) inertia index of G. Suppose that a connected graph G has at least one cycle and let g be the length of the shortest cycle in G. In this paper, we prove p(G) >= [g/2 ] -1. Moreover, the extremal graphs corresponding to p(G) = [g/2 ] - 1 and p(G) = [g/2 ] are completely characterized, respectively. (c) 2023 Elsevier Inc. All rights reserved.