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

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) >= inverted right perpendicularg/2inverted left perpendicular - 1. Furthermore, all extremal graphs corresponding to i(-) (G ) = inverted right perpendicularg/2inverted left perpendicular - 1 and i(-) (G ) = inverted right perpendicularg/2inverted left perpendicular are characterized, respectively. (c) 2024 Elsevier B.V. All rights reserved.