ABSTRACT

Let $G^{\varphi }$ be an n-vertex complex unit gain graph and let G be its underlying graph. The adjacency rank of $G^{\varphi }$, written as $r\left(G^{\varphi }\right)$, is the rank of its adjacency matrix and denote by ${\alpha }^{\prime }\left(G\right)$ the matching number of the underlying graph G. In this contribution, based on combinatorial interpretation of all the coefficients of the characteristic polynomial of $G^{\varphi }$, we determine sharp upper and lower bounds on $r\left(G^{\varphi }\right)-2{\alpha }^{\prime }\left(G\right).$ Furthermore, we establish sharp lower bounds on $r\left(G^{\varphi }\right)-{\alpha }^{\prime }\left(G\right)$ and $r\left(G^{\varphi }\right)/{\alpha }^{\prime }\left(G\right)$. All the corresponding extremal complex unit gain graphs are characterized.

中文翻译：

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复单位增益图的邻接秩与其底层图匹配数的关系

摘要

让$G^{\phi }$是一个n顶点复单位增益图，并让G是它的基础图。邻接等级$G^{\phi }$，写为$r\left(G^{\phi }\right)$, 是其邻接矩阵的秩，表示为${\alpha }^{\text{'}}\left(G\right)$底层图G的匹配数。在这个贡献中，基于对特征多项式的所有系数的组合解释$G^{\phi }$，我们确定尖锐的上限和下限$r\left(G^{\phi }\right)-2{\alpha }^{\text{'}}\left(G\right).$此外，我们在$r\left(G^{\phi }\right)-{\alpha }^{\text{'}}\left(G\right)$和$r\left(G^{\phi }\right)/{\alpha }^{\text{'}}\left(G\right)$. 所有相应的极值复单位增益图都被表征。