A Study on Eulerian Proper Power Graph for finite groups

Abstract

G be a finite group.   The power graph ℒ (G) is an undirected graph with vertex set G where two distinct vertices u and v in ℒ (G) are adjacent if u = v^α or v = u^β for some α, β in N. The proper power graph denoted by ℒ ^/(G) of the group G is the graph obtained by deleting the identity element from G. i.e, ℒ ^/ (G) = ℒ ^/ (G/e). In this paper, we Examine the Eulerian graph property for proper power graphs of certain finite groups such as S₃ , D₄, Q₈, D₅, A₄, D₆, Q₁₂, D₇.