Signature of power graphs

Abstract

The signature s(G) of a graph G is defined as the difference between its positive inertia index and the negative inertia index. In 2013, H. Ma et al. (2013) [8] conjectured that −c3(G)≤s(G)≤c5(G) for an arbitrary simple graph G, where ci(G) denotes the number of cycles in G with length i modulo 4. In 2014, L. Wang et al. [10] proved that −c3(Tk)≤s(Tk)≤c5(Tk) for any tree T and for any k≥2. In this paper, we prove that −c3(Gk)≤s(Gk)≤c5(Gk) for any simple graph G and for any k≥2, thus extend the main result of [10] to more general cases.