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Relation between the skew-rank of an oriented graph and the independence number of its underlying graph

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 Added by Shuchao Li
 Publication date 2017
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and research's language is English




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An oriented graph $G^sigma$ is a digraph without loops or multiple arcs whose underlying graph is $G$. Let $Sleft(G^sigmaright)$ be the skew-adjacency matrix of $G^sigma$ and $alpha(G)$ be the independence number of $G$. The rank of $S(G^sigma)$ is called the skew-rank of $G^sigma$, denoted by $sr(G^sigma)$. Wong et al. [European J. Combin. 54 (2016) 76-86] studied the relationship between the skew-rank of an oriented graph and the rank of its underlying graph. In this paper, the correlation involving the skew-rank, the independence number, and some other parameters are considered. First we show that $sr(G^sigma)+2alpha(G)geqslant 2|V_G|-2d(G)$, where $|V_G|$ is the order of $G$ and $d(G)$ is the dimension of cycle space of $G$. We also obtain sharp lower bounds for $sr(G^sigma)+alpha(G),, sr(G^sigma)-alpha(G)$, $sr(G^sigma)/alpha(G)$ and characterize all corresponding extremal graphs.



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A complex unit gain graph (or $mathbb{T}$-gain graph) is a triple $Phi=(G, mathbb{T}, varphi)$ ($(G, varphi)$ for short) consisting of a graph $G$ as the underlying graph of $(G, varphi)$, $mathbb{T}= { z in C:|z|=1 } $ is a subgroup of the multiplicative group of all nonzero complex numbers $mathbb{C}^{times}$ and a gain function $varphi: overrightarrow{E} rightarrow mathbb{T}$ such that $varphi(e_{ij})=varphi(e_{ji})^{-1}=overline{varphi(e_{ji})}$. In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph $(G, varphi)$ with order $n$, and prove that $2n-2c(G) leq r(G, varphi)+2alpha(G) leq 2n$. Where $r(G, varphi)$, $alpha(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $A(G, varphi)$, the independence number and the cyclomatic number of $G$, respectively. Furthermore, the properties of the complex unit gain graph that reaching the lower bound are characterized.
103 - Yuxuan Li 2020
We establish a lower bound for the energy of a complex unit gain graph in terms of the matching number of its underlying graph, and characterize all the complex unit gain graphs whose energy reaches this bound.
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