Common Neighborhood Energy of Commuting Graphs of Finite Groups

Rajat Kanti Nath, Walaa Nabil Taha Fasfous, Kinkar Chandra Das*, Yilun Shang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

The commuting graph of a finite non-abelian group G with center Z(G), denoted by Γc(G), is a simple undirected graph whose vertex set is G∖Z(G), and two distinct vertices x and y are adjacent if and only if xy=yx. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix CN(G) and the common neighborhood energy Ecn(G) of a simple graph G. A graph G is called CN-hyperenergetic if Ecn(G)>Ecn(Kn), where n=|V(G)| and Kn denotes the complete graph on n vertices. Two graphs G and H with equal number of vertices are called CN-equienergetic if Ecn(G)=Ecn(H). In this paper we compute the common neighborhood energy of Γc(G) for several classes of finite non-abelian groups, including the class of groups such that the central quotient is isomorphic to group of symmetries of a regular polygon, and conclude that these graphs are not CN-hyperenergetic. We shall also obtain some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic.
Original languageEnglish
Article number1651
Pages (from-to)1-12
Number of pages12
JournalSymmetry
Volume13
Issue number9
DOIs
Publication statusPublished - 8 Sep 2021

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