The cubic power graph of finite abelian groups

Abstract

Let G be a finite abelian group with identity 0. For an integer $n\ge 2,$ the additive power graph ${\Gamma }_{\mathit{apq}}(G)$ of G is the simple undirected graph with vertex set G in which two distinct vertices x and y are adjacent if and only if x + y = nt for some $t\in G$ with $nt\ne 0.$ When $n=2,$ the additive power graph has been studied in the name of square graph of finite abelian groups. In this paper, we study the additive power graph of G with n = 3 and name the graph as the cubic power graph. The cubic power graph of G is denoted by ${\Gamma }_{\mathit{cpg}}(G).$ More specifically, we obtain the diameter and the girth of the graph ${\Gamma }_{\mathit{cpg}}(G)$ and its complement ${\overline{\Gamma }}_{\mathit{cpg}}(G).$ Using these, we obtain a condition for ${\Gamma }_{\mathit{cpg}}(G)$ and its complement ${\overline{\Gamma }}_{\mathit{cpg}}(G)$ to be self-centered. Also, we obtain the independence number, the clique number and the chromatic number of ${\Gamma }_{\mathit{cpg}}(G)$ and its complement ${\overline{\Gamma }}_{\mathit{cpg}}(G)$ and hence we prove that ${\Gamma }_{\mathit{cpg}}(G)$ and its complement ${\overline{\Gamma }}_{\mathit{cpg}}(G)$ are weakly perfect. Also, we discuss about the perfectness of ${\Gamma }_{\mathit{cpg}}(G).$ At last, we obtain a condition for ${\Gamma }_{\mathit{cpg}}(G)$ and its complement ${\overline{\Gamma }}_{\mathit{cpg}}(G)$ to be vertex pancyclic.

Keywords:

• Abelian group
• additive power graph
• cubic power graph
• complement
• perfect
• chromatic number

2000 Mathematics Subject Classification:

• 05C25
• 05C40
• 05C69

Additional information

Funding

This research work is supported by CSIR Emeritus Scientist Scheme (No. 21 (1123)/20/EMR-II) of Council of Scientific and Industrial Research, Government of India through the second author.