arXiv subjects

Liangliang Zhai

Publications and source records attributed to Liangliang Zhai.

3 recordsLinked to original sources

arXiv2026

On finite groups whose coprime graph is a divisor graph

In this paper, we first characterize which generalized lexicographic products are divisor graphs. As applications, we show that power graphs, reduced power graphs and order graphs are all divisor graphs, which also implies the main result in [Power graph of a finite group is always divisor graph, Asian-European Journal of Mathematics 16 (2023)]. We then show that, the coprime graph of a group is a generalized lexicographic product, and characterize which coprime graphs are divisor graphs. Finally, we classify the finite groups $G$ having at most four prime divisors, whose coprime graphs are divisor graphs, and we also classify the finite groups $G$ whose coprime graphs are divisor graphs, if $G$ is a nilpotent group, a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, a direct product of two non-trivial groups, and a sporadic simple group.

math.GR↗

arXiv2024

(Total) Perfect codes in (extended) subgroup sum graphs

Given a finite group $G$ with identity $e$ and a normal subgroup $H$ of $G$, the subgroup sum graph $Γ_{G,H}$ (resp. extended subgroup sum graph $Γ_{G,H}^+$) of $G$ with respect to $H$ is the graph with vertex set $G$, in which distinct vertices $x$ and $y$ are adjacent whenever $xy\in H\setminus \{e\}$ (resp. $xy\in H$). A group $G$ is said to be {\em code-perfect} if for any normal subgroup $H$ of $G$, $Γ_{G,H}$ admits a perfect code. In this paper, we give a necessary and sufficient condition for which normal subgroups $H$ of $G$ satisfy that a (extended) subgroup sum graph of $G$ with respect to $H$ admits a (total) perfect code, and classify all code-perfect Dedekind groups. As an application, we classify all normal subgroups such that the subgroup sum graph of a cyclic group, a dihedral group or a dicyclic group with respect to such a normal subgroup admits perfect codes, respectively. We also determine all abelian groups $A$ and subgroups $H$ of $A$ such that $Γ_{A,H}$ admits a total perfect code.

math.CO↗

arXiv2021

Strong metric dimensions for power graphs of finite groups

Let $G$ be a finite group. The order supergraph of $G$ is the graph with vertex set $G$, and two distinct vertices $x,y$ are adjacent if $o(x)\mid o(y)$ or $o(y)\mid o(x)$. The enhanced power graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices are adjacent if they generate a cyclic subgroup. The reduced power graph of $G$ is the graph with vertex set $G$, and two distinct vertices $x,y$ are adjacent if $\langle x\rangle \subset \langle y\rangle$ or $\langle y\rangle \subset \langle x\rangle$. In this paper, we characterize the strong metric dimension of the order supergraph, the enhanced power graph and the reduced power graph of a finite group.

math.CO↗