Characterizing the negative inertia index of connected graphs in terms of their girth

Abstract

Suppose that a connected graph G has at least one cycle and let g be the length of the shortest cycle in G, which is called the girth of G. In this paper, we consider relationship between the girth of G and the number of negative eigenvalues (including multiplicities) of the adjacency matrix of G, known as negative inertia index of G and denoted by i−(G). We prove that i−(G)≥⌈g2⌉−1. Furthermore, all extremal graphs corresponding to i−(G)=⌈g2⌉−1 and i−(G)=⌈g2⌉ are characterized, respectively.