arXiv subjects

Aimei Yu

Publications and source records attributed to Aimei Yu.

3 recordsLinked to original sources

arXiv2019

Bounds for the rank of a complex unit gain graph in terms of the independence number

A complex unit gain graph (or $\mathbb{T}$-gain graph) is a triple $Φ=(G, \mathbb{T}, φ)$ ($(G, φ)$ for short) consisting of a graph $G$ as the underlying graph of $(G, φ)$, $\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 $φ: \overrightarrow{E} \rightarrow \mathbb{T}$ such that $φ(e_{ij})=φ(e_{ji})^{-1}=\overline{φ(e_{ji})}$. In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph $(G, φ)$ with order $n$, and prove that $2n-2c(G) \leq r(G, φ)+2α(G) \leq 2n$. Where $r(G, φ)$, $α(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $A(G, φ)$, 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.

math.CO↗

arXiv2019

On the inertia index of a mixed graph with the matching number

A mixed graph $\widetilde{G}$ is obtained by orienting some edges of $G$, where $G$ is the underlying graph of $\widetilde{G}$. The positive inertia index, denoted by $p^{+}(G)$, and the negative inertia index, denoted by $n^{-}(G)$, of a mixed graph $\widetilde{G}$ are the integers specifying the numbers of positive and negative eigenvalues of the Hermitian adjacent matrix of $\widetilde{G}$, respectively. In this paper, we study the positive and negative inertia index of the mixed unicyclic graph. Moreover, we give the upper and lower bounds of the positive and negative inertia index of the mixed graph, and characterize the mixed graphs which attain the upper and lower bounds respectively.

math.CO↗

arXiv2017

On extremal cacti with respect to the edge Szeged index and edge-vertex Szeged index

The edge Szeged index and edge-vertex Szeged index of a graph are defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$ and $Sz_{ev}(G)=\frac{1}{2} \sum\limits_{uv \in E(G)}[n_{u}(uv|G)m_{v}(uv|G)+n_{v}(uv|G)m_{u}(uv|G)],$ respectively, where $m_{u}(uv|G)$ (resp., $m_{v}(uv|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $n_{u}(uv|G)$ (resp., $n_{v}(uv|G)$) is the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), respectively. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, the lower bounds of edge Szeged index and edge-vertex Szeged index for cacti with order $n$ and $k$ cycles are determined, and all the graphs that achieve the lower bounds are identified.

math.CO↗