Normal Subgroup Based Power Graph of a finite Group

2016/01/18 by A. K. Bhuniya, Bhuniya, A. K., Sudip Bera +1
Mathematics · Engineering · #Finite Group Theory Research #Rings, Modules, and Algebras #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1601.04431

Abstract

For a finite group G with a normal subgroup H, the normal subgroup based power graph of G, denoted by Γ_H(G) whose vertex set V(Γ_H(G))=(G∖ H)\bigcup \e\ and two vertices a and b are edge connected if aH=b^mH or bH=aⁿH for some m, n ∈ ℕ. In this paper we obtain some fundamental characterizations of the normal subgroup based power graph. We show some relation between the graph Γ_H(G) and the power graph Γ((G)/(H)). We show that Γ_H(G) is complete if and only of (G)/(H) is cyclic group of order 1 or p^m, where p is prime number and m∈ ℕ. Γ_H(G) is planar if and only if |H|=2 or 3 and (G)/(H)≅ ℤ₂× ℤ₂ × ⋯ × ℤ₂. Also Γ_H(G) is Eulerian if and only if |G|≡ |H| mod 2.

Citations

• The Zero-Divisor Graph of a Commutative Ring
• Coloring of commutative rings

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• The enhanced quotient graph of the quotient of a finite group
• On the power graph of a finite group
• Generalized power graph of groups
• Power Graphs of Finite Groups