The enhanced quotient graph of the quotient of a finite group

Abstract

For a finite group G with a normal subgroup H, the enhanced quotient graph of G/H, denoted by G H (G), is the graph with vertex set V = (G\H) ∪ {e} and two vertices x and y are edge connected if xH = yH or xH, yH ∈ zH for some z ∈ G. In this article, we characterize the enhanced quotient graph of G/H. The graph G H (G) is complete if and only if G/H is cyclic, and G H (G) is Eulerian if and only if |G/H| is odd. We show some relation between the graph G H (G) and the enhanced power graph G(G/H) that was introduced by Sudip Bera and A.K. Bhuniya (2016). The graph G H (G) is complete if and only if G/H is cyclic if and only if G(G/H) is complete. The graph G H (G) is Eulerian if and only if |G| is odd if and only if G(G) is Eulerian, i.e., the property of being Eulerian does not depend on the normal subgroup H.

On The Enhanced Power Graph of a Semigroup

“…In [5], Bera et al characterized the abelian groups and the non abelian p-groups having dominatable enhanced power graphs. In [13], Dupont et al determined the rainbow connection number of enhanced power graph of a finite group G. Later, Dupont et al studied the graph theoretic properties in [12] of enhanced quotient graph of a finite group G. Ma et al [20] investigated the metric dimension of an enhanced power graph of finite groups. Zahirović et al [26] proved that two finite abelian groups are isomorphic if their enhanced power graphs are isomorphic.…”

Equality of various graphs on finite semigroups

“…If these two graphs of G do not coincide, then to measure how much the power graph is close to the commuting graph of G, they introduced a new graph so called enhanced power graph of a group G. The enhanced power graph of a group G is the simple graph whose vertex set is the group G and two distinct vertices x, y are adjacent if x, y ∈ z for some z ∈ G. In [7], Bera et al studied the enhanced power graph of finite groups. Daniel et al [16] studied graph theoretic properties (connectivity, completeness etc.) of the enhanced power graph of the quotient group G/H.…”