[Paper Review] Normal Subgroup Based Power Graph of a finite Group

A. K. Bhuniya, Sudip Bera|arXiv (Cornell University)|Jan 18, 2016

Finite Group Theory Research8 references3 citations

TL;DR

This paper introduces the normal subgroup based power graph Γₕ(G) of a finite group G with a normal subgroup H, defining edges based on coset power relations. It establishes key connections between graph properties of Γₕ(G) and group properties of G/H, proving that Γₕ(G) is complete iff G/H is trivial or of order p^m, planar iff |H| ∈ {2,3} and G/H is elementary abelian 2-group, and Eulerian iff |G| ≡ |H| mod 2.

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 $V(Γ_H(G))=(G\setminus H)\bigcup \{e\}$ and two vertices $a$ and $b$ are edge connected if $aH=b^mH$ or $bH=a^nH$ for some $m, n \in \mathbb{N}$. 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 $Γ(\frac{G}{H})$. We show that $Γ_H(G)$ is complete if and only of $\frac{G}{H}$ is cyclic group of order $1$ or $p^m$, where $p$ is prime number and $m\in \mathbb{N}$. $Γ_H(G)$ is planar if and only if $|H|=2$ or $3$ and $\frac{G}{H}\cong \mathbb{Z}_2 imes \mathbb{Z}_2 imes \cdots imes \mathbb{Z}_2$. Also $Γ_H(G)$ is Eulerian if and only if $|G|\equiv |H|$ mod$ 2$.

Motivation & Objective

• To generalize the concept of power graphs by introducing a new graph construction based on normal subgroups.
• To investigate how graph-theoretic properties of Γₕ(G) relate to group-theoretic properties of the quotient group G/H.
• To characterize fundamental properties of Γₕ(G), including completeness, planarity, Eulerian nature, and connectivity.
• To compute key invariants such as clique number, chromatic number, and vertex connectivity of Γₕ(G).

Proposed method

• Define the normal subgroup based power graph Γₕ(G) with vertex set (G \ H) ∪ {e}, where e is the identity.
• Establish adjacency via coset power relations: a ~ b iff aH = b^m H or bH = a^n H for some m,n ∈ ℕ.
• Use quotient group G/H to analyze structural properties of Γₕ(G), particularly through isomorphisms and coset dynamics.
• Prove that Γₕ(G) is a Cayley graph iff G/H is a cyclic p-group.
• Derive formulas for edge count, clique number w(Γₕ(G)) = |H|(M−1)+1 where M = w(Γ(G/H)), and chromatic number χ(Γₕ(G)) = w(Γₕ(G)).
• Use vertex connectivity arguments involving deletion of entire cosets to show κ(Γₕ(G)) = |H|(k−1)+1 where k = κ(Γ(G/H)).

Experimental results

Research questions

• RQ1 When is the normal subgroup based power graph Γₕ(G) complete, and what group-theoretic condition on G/H ensures this?
• RQ2 Under what conditions is Γₕ(G) planar, and how does the structure of H and G/H affect this?
• RQ3 What is the necessary and sufficient condition for Γₕ(G) to be Eulerian?
• RQ4 How do the clique number and chromatic number of Γₕ(G) relate to those of the power graph of G/H?
• RQ5 What is the vertex connectivity of Γₕ(G), and how is it determined by the connectivity of Γ(G/H)?

Key findings

• Γₕ(G) is complete if and only if G/H is trivial or of order p^m for some prime p and m ∈ ℕ.
• Γₕ(G) is planar if and only if |H| ∈ {2, 3} and G/H ≅ ℤ₂ × ℤ₂ × ⋯ × ℤ₂.
• Γₕ(G) is Eulerian if and only if |G| ≡ |H| (mod 2).
• The clique number of Γₕ(G) is w(Γₕ(G)) = |H|(M−1)+1, where M = w(Γ(G/H)).
• The chromatic number of Γₕ(G) equals its clique number, so χ(Γₕ(G)) = |H|(M−1)+1, proving Γₕ(G) is perfect.
• The vertex connectivity of Γₕ(G) is |H|(k−1)+1, where k = κ(Γ(G/H)).