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Register a new userThe rank of a complex unit gain graph in terms of the matching number
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A complex unit gain graph (or ${mathbb T}$-gain graph) is a triple $Phi=(G, {mathbb T}, varphi)$ (or $(G, varphi)$ for short) consisting of a simple graph $G$, as the underlying graph of $(G, varphi)$, the set of unit complex numbers $mathbb{T}= { z in C:|z|=1 }$ and a gain function $varphi: overrightarrow{E} rightarrow mathbb{T}$ with the property that $varphi(e_{i,j})=varphi(e_{j,i})^{-1}$. In this paper, we prove that $2m(G)-2c(G) leq r(G, varphi) leq 2m(G)+c(G)$, where $r(G, varphi)$, $m(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $H(G, varphi)$, the matching number and the cyclomatic number of $G$, respectively. Furthermore, the complex unit gain graphs $(G, mathbb{T}, varphi)$ with $r(G, varphi)=2m(G)-2c(G)$ and $r(G, varphi)=2m(G)+c(G)$ are characterized. These results generalize the corresponding known results about undirected graphs, mixed graphs and signed graphs. Moreover, we show that $2m(G-V_{0}) leq r(G, varphi) leq 2m(G)+b(G)$ holds for any subset $V_0$ of $V(G)$ such that $G-V_0$ is acyclic and $b(G)$ is the minimum integer $|S|$ such that $G-S$ is bipartite for $S subset V(G)$.
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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.
Let $Phi=(G, varphi)$ be a complex unit gain graph (or $mathbb{T}$-gain graph) and $A(Phi)$ be its adjacency matrix, where $G$ is called the underlying graph of $Phi$. The rank of $Phi$, denoted by $r(Phi)$, is the rank of $A(Phi)$. Denote by $theta(G)=|E(G)|-|V(G)|+omega(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $omega(G)$ are the number of edges, the number of vertices and the number of connected components of $G$, respectively. In this paper, we investigate bounds for $r(Phi)$ in terms of $r(G)$, that is, $r(G)-2theta(G)leq r(Phi)leq r(G)+2theta(G)$, where $r(G)$ is the rank of $G$. As an application, we also prove that $1-theta(G)leqfrac{r(Phi)}{r(G)}leq1+theta(G)$. All corresponding extremal graphs are characterized.
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.
A complex unit gain graph is a graph where each orientation of an edge is given a complex unit, which is the inverse of the complex unit assigned to the opposite orientation. We extend some fundamental concepts from spectral graph theory to complex unit gain graphs. We define the adjacency, incidence and Laplacian matrices, and study each of them. The main results of the paper are eigenvalue bounds for the adjacency and Laplacian matrices.
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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