The Signature of Chordal Graphs and Cographs

The signature s(G) of a graph G is defined as the difference between its positive inertia index and negative inertia index. In 2013, Ma et al. conjectured that $$-c_3(G)\le s(G) \le c_5(G)$$ for an arbitrary simple graph G, where $$c_3(G)$$ denotes the number of cycles in G of length 3 modulo 4, $$c_5(G)$$ denotes the number of cycles in G of length 1 modulo 4. In this paper, we give some inequalities of signature of graphs, as applications, we prove that this conjecture holds for chordal graphs and cographs.