[Paper Review] The enhanced quotient graph of the quotient of a finite group

Luis A. Dupont, Daniel G. Mendoza+1 more|arXiv (Cornell University)|Jul 4, 2017

Finite Group Theory Research6 references3 citations

TL;DR

This paper introduces and characterizes the enhanced quotient graph $\mathcal{G}_H(G)$ of a finite group $G$ modulo a normal subgroup $H$, defining it via coset relations and adjacency based on cyclic subgroup generation. The key contribution is establishing that $\mathcal{G}_H(G)$ is complete if and only if $G/H$ is cyclic, and Eulerian if and only if $|G/H|$ is odd, with strong structural parallels to the enhanced power graph $\mathcal{G}(G/H)$, including clique structure and planarity conditions.

ABSTRACT

For a finite group $G$ with a normal subgroup $H$, the enhanced quotient graph of $G/H$, denoted by $\mathcal{G}_{H}(G),$ is the graph with vertex set $V=(G\backslash H)\cup \{e\}$ and two vertices $x$ and $y$ are edge connected if $xH = yH$ or $xH,yH\in \langle zH angle$ for some $z\in G$. In this article, we characterize the enhanced quotient graph of $G/H$. The graph $\mathcal{G}_{H}(G)$ is complete if and only if $G/H$ is cyclic, and $\mathcal{G}_{H}(G)$ is Eulerian if and only if $|G/H|$ is odd. We show some relation between the graph $\mathcal{G}_{H}(G)$ and the enhanced power graph $\mathcal{G}(G/H)$ that was introduced by Sudip Bera and A.K. Bhuniya (2016). The graph $\mathcal{G}_H(G)$ is complete if and only if $G/H$ is cyclic if and only if $\mathcal{G}(G/H)$ is complete. The graph $\mathcal{G}_H(G)$ is Eulerian if and only if $|G|$ is odd if and only if $\mathcal{G}(G)$ is Eulerian, i.e., the property of being Eulerian does not depend on the normal subgroup $H$.

Motivation & Objective

• To define and analyze the enhanced quotient graph $\mathcal{G}_H(G)$, a novel graph construction on the quotient group $G/H$ using coset-based adjacency.
• To establish structural properties of $\mathcal{G}_H(G)$, including connectivity, clique structure, and Eulerian and planar characteristics.
• To compare $\mathcal{G}_H(G)$ with the enhanced power graph $\mathcal{G}(G/H)$, revealing deep algebraic-graph-theoretic equivalences.
• To investigate the conditions under which $\mathcal{G}_H(G)$ is Eulerian, complete, or planar, linking these to group-theoretic invariants of $G/H$.
• To extend results on the deleted enhanced quotient graph $\mathcal{G}_H^*(G)$, analyzing its connectivity and structural properties in relation to the center and element orders of $G/H$.

Proposed method

• Define $\mathcal{G}_H(G)$ with vertex set $(G \setminus H) \cup \{e\}$, where edges exist if vertices lie in the same coset or are contained in a cyclic subgroup of $G/H$.
• Use the condition $\{a,b\} \in E(\mathcal{G}_H(G))$ iff $aH = bH$ or $\{aH, bH\} \in E(\mathcal{G}(G/H))$ to link $\mathcal{G}_H(G)$ to the enhanced power graph $\mathcal{G}(G/H)$.
• Apply group-theoretic results on cyclic subgroups and element orders to derive clique number $\omega(\mathcal{G}_H(G)) = |H|^{s-1} + 1$, where $s$ is the maximal order of a cyclic subgroup in $G/H$.
• Leverage Kuratowski's theorem to determine planarity: $\mathcal{G}_H(G)$ is non-planar iff $|H| \geq 4$, or $|H|=1$ and $s \geq 5$, or $|H|=2$ and $s \geq 3$, or $|H|=3$ and $s \geq 2$.
• Analyze the deleted graph $\mathcal{G}_H^*(G)$ by removing the identity vertex, and derive connectivity conditions based on the uniqueness of minimal subgroups and center structure in $G/H$.
• Use the equivalence of $\mathcal{G}_H(G)$ being $k$-partite to the condition that $|aH| \leq 2$ for all $aH \in G/H$, leading to $G/H \cong \mathbb{Z}_2^k$ when the circumference is $|H|$.

Experimental results

Research questions

• RQ1 When is the enhanced quotient graph $\mathcal{G}_H(G)$ complete, and how does this relate to the cyclicity of $G/H$?
• RQ2 Under what conditions is $\mathcal{G}_H(G)$ Eulerian, and does this property depend on the normal subgroup $H$?
• RQ3 How does the structure of $\mathcal{G}_H(G)$ relate to the enhanced power graph $\mathcal{G}(G/H)$ in terms of completeness and Eulerian properties?
• RQ4 What are the necessary and sufficient conditions for $\mathcal{G}_H(G)$ to be planar, and how do the size of $H$ and the exponent of $G/H$ affect this?
• RQ5 What characterizations hold for the deleted enhanced quotient graph $\mathcal{G}_H^*(G)$, particularly regarding its $k$-partiteness and circumference?

Key findings

• $\mathcal{G}_H(G)$ is complete if and only if $G/H$ is cyclic, and this condition is equivalent to the enhanced power graph $\mathcal{G}(G/H)$ being complete.
• $\mathcal{G}_H(G)$ is Eulerian if and only if $|G/H|$ is odd, and this property is independent of the choice of normal subgroup $H$.
• The clique number of $\mathcal{G}_H(G)$ is $|H|^{s-1} + 1$, where $s$ is the maximum order of a cyclic subgroup in $G/H$.
• $\mathcal{G}_H(G)$ contains at least $|H|^{[G:H]-1}$ subgraphs isomorphic to $\mathcal{G}(G/H)$, reflecting the coset-based construction.
• The graph $\mathcal{G}_H(G)$ is non-planar if and only if $|H| \geq 4$, or $|H|=1$ and $s \geq 5$, or $|H|=2$ and $s \geq 3$, or $|H|=3$ and $s \geq 2$.
• The deleted enhanced quotient graph $\mathcal{G}_H^*(G)$ is $k$-partite with part size 1 per coset if and only if $G/H \cong \mathbb{Z}_2^k$, and this holds precisely when the circumference of $\mathcal{G}_H^*(G)$ is $|H|$.