On the enhanced power graph of a finite group. Issue 4 (3rd April 2021)

• Record Type: Journal Article
• Authors: Panda, Ramesh Prasad; Dalal, Sandeep; Kumar, Jitender
• Is Part Of: Communications in algebra. Volume 49:Issue 4(2021)
• Page Start: 1697
• Page End: 1716
• Publication Date: 2021-04-03
• Subjects: Enhanced power graph -- finite groups -- independence number -- matching -- minimum degree; 05C25; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1847289
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: The enhanced power graph P e ( G ) of a group G is a graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number, and matching number of enhanced power graphs of finite groups. We first study these graph invariants for P e ( G ) when G is any finite group and then determine them when G is a finite abelian p -group, U 6 n = 〈 a, b : a 2 n = b 3 = e, b a = a b − 1 〉, the dihedral group D 2 n, or the semidihedral group S D 8 n . If G is any of these groups, we prove that P e ( G ) is perfect and then obtain its strong metric dimension. Additionally, we give an expression for the independence number of P e ( G ) for any finite abelian group G . These results along with certain known equalities yield the edge connectivity, vertex covering number, and edge covering number of enhanced power graphs of the respective groups as well.