The signature of two generalizations of line graphs

Abstract

The signature s(G) of a graph G is defined as the difference between its positive inertia index and negative inertia index. In 2013, H. Ma et al. ([1]) conjectured that −c3(G)≤s(G)≤c5(G) for an arbitrary simple graph G, where c3(G) denotes the number of cycles in G of length 3 modulo 4, c5(G) denotes the number of cycles in G of length 1 modulo 4. In this paper, we prove that this conjecture holds for claw-free graphs and graphs with least eigenvalue ≥−2.