LAUSR

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 − c 3 ( G ) ≤ s ( G ) ≤ c 5 ( G ) for an arbitrary simple graph G , where c i ( G ) denotes the number of cycles in G with length i modulo 4. In 2014, L. Wang et al. [10] proved that − c 3 ( T k ) ≤ s ( T k ) ≤ c 5 ( T k ) for any tree T and for any k ≥ 2 . In this paper, we prove that − c 3 ( G k ) ≤ s ( G k ) ≤ c 5 ( G k ) for any simple graph G and for any k ≥ 2 , thus extend the main result of [10] to more general cases.