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

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 $n.$