Common Neighborhood Energy of Commuting Graphs of Finite Groups

Symmetry

卷:13

Yilun Shang¹  Walaa Nabil Taha Fasfous²  Rajat Kanti Nath²  Kinkar Chandra Das³ 

[1] Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK;

[2] Department of Mathematical Sciences, Tezpur University, Tezpur 784028, India;

[3] Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea;

关键词: commuting graph;    CN-energy;    finite group;   

DOI  :  10.3390/sym13091651

来源: DOAJ

【 摘 要 】

The commuting graph of a finite non-abelian group G with center $Z\left(G\right)$, denoted by ${\Gamma }_c\left(G\right)$, is a simple undirected graph whose vertex set is $G\setminus Z\left(G\right)$, 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 $CN\left(\mathcal{G}\right)$ and the common neighborhood energy $E_{cn}\left(\mathcal{G}\right)$ of a simple graph $\mathcal{G}$. A graph $\mathcal{G}$ is called CN-hyperenergetic if $E_{cn}\left(\mathcal{G}\right)>E_{cn}\left(K_n\right)$, where $n=\left|V\right(\mathcal{G}\left)\right|$ and $K_n$ denotes the complete graph on n vertices. Two graphs $\mathcal{G}$ and $\mathcal{H}$ with equal number of vertices are called CN-equienergetic if $E_{cn}\left(\mathcal{G}\right)=E_{cn}\left(\mathcal{H}\right)$. In this paper we compute the common neighborhood energy of ${\Gamma }_c\left(G\right)$ 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.

【 授权许可】

Unknown