On the strong domination number of proper enhanced power graphs of finite groups

Abstract

The enhanced power graph of a group G is a graph with vertex set G, where two distinct vertices \(\mathbb{x}\) and \(\mathbb{y}\) are adjacent if and only if there exists an element \(\mathbb{w}\) in G such that both \(\mathbb{x}\) and \(\mathbb{y}\) are powers of \(\mathbb{w}\). To obtain the proper enhanced power graph, we consider the induced subgraph on the set \(G \setminus D\), where D represents the set of dominating vertices in the enhanced power graph. In this paper, we aim to determine the strong domination number of the proper enhanced power graphs of finite nilpotent groups.