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Register a new userThe relation between the independence number and rank of a signed graph
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A signed graph $(G, sigma)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $(G, sigma)$. Let $c(G)$, $alpha(G)$ and $r(G, sigma)$ be the cyclomatic number, the independence number and the rank of the adjacency matrix of $(G, sigma)$, respectively. In this paper, we study the relation among the independence number, the rank and the cyclomatic number of a signed graph $(G, sigma)$ with order $n$, and prove that $2n-2c(G) leq r(G, sigma)+2alpha(G) leq 2n$. Furthermore, the signed graphs that reaching the lower bound are investigated.
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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.
Given a simple graph $G=(V_G, E_G)$ with vertex set $V_G$ and edge set $E_G$, the mixed graph $widetilde{G}$ is obtained from $G$ by orienting some of its edges. Let $H(widetilde{G})$ denote the Hermitian adjacency matrix of $widetilde{G}$ and $A(G)$ be the adjacency matrix of $G$. The $H$-rank (resp. rank) of $widetilde{G}$ (resp. $G$), written as $rk(widetilde{G})$ (resp. $r(G)$), is the rank of $H(widetilde{G})$ (resp. $A(G)$). Denote by $d(G)$ the dimension of cycle spaces of $G$, that is $d(G) = |E_G|-|V_G|+omega(G)$, where $omega(G),$ denotes the number of connected components of $G$. In this paper, we concentrate on the relation between the $H$-rank of $widetilde{G}$ and the rank of $G$. We first show that $-2d(G)leqslant rk(widetilde{G})-r(G)leqslant 2d(G)$ for every mixed graph $widetilde{G}$. Then we characterize all the mixed graphs that attain the above lower (resp. upper) bound. By these obtained results in the current paper, all the main results obtained in cite{004,1} may be deduced consequently.
In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $phicolon V(G)to mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $phi(u)$ is different from the colour $sigma(uv)phi(v)$, where is $sigma(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavskys research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks theorem to signed graphs.
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.
An edge-colored connected graph $G$ is properly connected if between every pair of distinct vertices, there exists a path that no two adjacent edges have a same color. Fujita (2019) introduced the optimal proper connection number ${mathrm{pc}_{mathrm{opt}}}(G)$ for a monochromatic connected graph $G$, to make a connected graph properly connected efficiently. More precisely, ${mathrm{pc}_{mathrm{opt}}}(G)$ is the smallest integer $p+q$ when one converts a given monochromatic graph $G$ into a properly connected graph by recoloring $p$ edges with $q$ colors. In this paper, we show that ${mathrm{pc}_{mathrm{opt}}}(G)$ has an upper bound in terms of the independence number $alpha(G)$. Namely, we prove that for a connected graph $G$, ${mathrm{pc}_{mathrm{opt}}}(G)le frac{5alpha(G)-1}{2}$. Moreoevr, for the case $alpha(G)leq 3$, we improve the upper bound to $4$, which is tight.
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