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.

中文翻译：

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和弦图和图形的签名

曲线G的签名s（G）被定义为其正惯性指数和负惯性指数之间的差。2013年，Ma等。推测\（-c_3（G）\ le s（G）\ le c_5（G）\）对于任意简单图G，其中\（c_3（G）\）表示长度为3模的G中的循环数4，\（c_5（G）\）表示长度为1模4的G中的循环数。。