On the super graphs and reduced super graphs of some finite groups

TL;DR

This paper introduces and characterizes super graphs and reduced super graphs of finite groups, focusing on their structure, dominant vertices, and connectivity properties, especially for symmetric and alternating groups.

Contribution

It provides new characterizations of when certain super graphs are equal, identifies dominant vertices in order super commuting graphs, and analyzes connectivity and diameter for reduced super graphs of symmetric and alternating groups.

Findings

Identity element is the only dominant vertex in $ ext{Delta}^o(S_n)$ and $ ext{Delta}^o(A_n)$ for $n \\geq 4$.

Connectivity of reduced order super commuting graphs depends on $n$, with many being connected.

Diameter of connected reduced super graphs is at most 3, with many cases achieving diameter 3.

Abstract

For a finite group $G$, let $B$ be an equivalence (equality, conjugacy or order) relation on $G$ and let $A$ be a (power, enhanced power or commuting) graph with vertex set $G$. The $B$ super $A$ graph is a simple graph with vertex set $G$ and two vertices are adjacent if either they are in the same $B$-equivalence class or there are elements in their $B$-equivalence classes that are adjacent in the original $A$ graph. The graph obtained by deleting the dominant vertices (adjacent to all other vertices) from a $B$ super $A$ graph is called the reduced $B$ super $A$ graph. In this article, for some pairs of $B$ super $A$ graphs, we characterize the finite groups for which a pair of graphs are equal. We also characterize the dominant vertices for the order super commuting graph $\Delta^o(G)$ of $G$ and prove that for $n\geq 4$ the identity element is the only dominant vertex of $\Delta^o(S_n)$ and $\Delta^o(A_n)$. We characterize the values of $n$ for which the reduced order super commuting graph $\Delta^o(S_n)^*$ of $S_n$ and the reduced order super commuting graph $\Delta^o(A_n)^*$ of $A_n$ are connected. We also prove that if $\Delta^o(S_n)^*$ (or $\Delta^o(A_n)^*$) is connected then the diameter is at most $3$ and shown that the diameter is $3$ for many value of $n.$