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

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 Delta(0)(G) of G and prove that for n >= 4 the identity element is the only dominant vertex of Delta(0)(S-n) and Delta(0)(A(n)). We characterize the values of n for which the reduced order super commuting graph Delta(0)(S-n)* of S-n and the reduced order super commuting graph Delta(0)(A(n))* of A(n) are connected. We also prove that if Delta(0)(S-n)* (or Delta(0)(A(n))*) is connected then the diameter is at most 3 and show that the diameter is 3 for many values of n. (c) 2023 Elsevier B.V. All rights reserved.