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 simple (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 distinct 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 Δo(G) of G and prove that for n≥4 the identity element is the only dominant vertex of Δo(Sn) and Δo(An). We characterize the values of n for which the reduced order super commuting graph Δo(Sn)⁎ of Sn and the reduced order super commuting graph Δo(An)⁎ of An are connected. We also prove that if Δo(Sn)⁎ (or Δo(An)⁎) is connected then the diameter is at most 3 and show that the diameter is 3 for many values of n.