Searcharxiv⌕ Search

arXiv subjects

Xuanlong Ma

Publications and source records attributed to Xuanlong Ma.

29 records · Page 2Linked to original sources
arXiv2017

The power index of a graph

The {\em power index} $Θ(Γ)$ of a graph $Γ$ is the least order of a group $G$ such that $Γ$ can embed into the power graph of $G$. Furthermore, this group $G$ is {\em $Γ$-optimal} if $G$ has order $Θ(Γ)$. We say that $Γ$ is {\em power-critical} if its order equals to $Θ(Γ)$. This paper focuses on the power indices of complete graphs, complete bipartite graphs and $1$-factors. We classify all power-critical graphs $Γ'$ in these three families, and give a necessary and sufficient condition for $Γ'$-optimal groups.

math.CO↗
arXiv2016

Finite groups with star-free noncyclic graphs

For a finite noncyclic group $G$, let $\Cyc(G)$ be a set of elements $a$ of $G$ such that $\langle a,b\rangle$ is cyclic for each $b$ of $G$. The noncyclic graph of $G$ is a graph with the vertex set $G\setminus \Cyc(G)$, having an edge between two distinct vertices $x$ and $y$ if $\langle x, y\rangle$ is not cyclic. In this paper, we classify all finite noncyclic groups whose noncyclic graphs are $K_{1,n}$-free, where $3\leq n\leq 6$.

math.GR↗
arXiv2015

Power graphs of (non)orientable genus two

The power graph $Γ_G$ of a finite group $G$ is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this paper, we classify the finite groups whose power graphs have (non)orientable genus two.

math.GR↗
arXiv2015

Non-cyclic graphs of (non)orientable genus one

Let $G$ be a finite non-cyclic group. The non-cyclic graph $Γ_G$ of $G$ is the graph whose vertex set is $G\setminus Cyc(G)$, two distinct vertices being adjacent if they do not generate a cyclic subgroup, where $Cyc(G)=\{a\in G: \langle a,b\rangle\ \text{is cyclic for each } b\in G\}$. In this paper, we classify all finite non-cyclic groups $G$ such that $Γ_G$ has (non)orientable genus one.

math.GR↗
arXiv2015

The rainbow connection number of the power graph of a finite group

This paper studies the rainbow connection number of the power graph $Γ_G$ of a finite group $G$. We determine the rainbow connection number of $Γ_G$ if $G$ has maximal involutions or is nilpotent, and show that the rainbow connection number of $Γ_G$ is at most three if $G$ has no maximal involutions. The rainbow connection numbers of power graphs of some nonnilpotent groups are also given.

math.CO↗
arXiv2015

On finite groups all of whose cubic Cayley graphs are integral

For any positive integer $k$, let $\mathcal{G}_k$ denote the set of finite groups $G$ such that all Cayley graphs ${\rm Cay}(G,S)$ are integral whenever $|S|\le k$. Est${\rm \acute{e}}$lyi and Kov${\rm \acute{a}}$cs \cite{EK14} classified $\mathcal{G}_k$ for each $k\ge 4$. In this paper, we characterize the finite groups each of whose cubic Cayley graphs is integral. Moreover, the class $\mathcal{G}_3$ is characterized. As an application, the classification of $\mathcal{G}_k$ is obtained again, where $k\ge 4$.

math.GR↗
arXiv2015

Rainbow connectivity of the non-commuting graph of a finite group

Let $G$ be a finite non-abelian group. The non-commuting graph $Γ_G$ of $G$ has the vertex set $G\setminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent if $xy\ne yx$, where $Z(G)$ is the center of $G$. We prove that the rainbow $2$-connectivity of $Γ_G$ is $2$. In particular, the rainbow connection number of $Γ_G$ is $2$. Moreover, for any positive integer $k$, we prove that there exist infinitely many non-abelian groups $G$ such that the rainbow $k$-connectivity of $Γ_G$ is $2$.

math.CO↗
arXiv2014

On the power graph of a finite group

The power graph $\mathcal P_G$ of a finite group $G$ is the graph with the vertex set $G$, where two elements are adjacent if one is a power of the other. We first show that $\mathcal P_G$ has an transitive orientation, so it is a perfect graph and its core is a complete graph. Then we use the poset on all cyclic subgroups (under usual inclusion) to characterise the structure of $\mathcal P_G$. Finally, the closed formula for the metric dimension of $\mathcal P_G$ is established. As an application, we compute the metric dimension of the power graph of a cyclic group.

math.CO↗