[Paper Review] Energy of commuting graph of finite groups whose centralizers are Abelian

Reza Sharafdini, Rajat Kanti Nath+1 more|arXiv (Cornell University)|Apr 21, 2017

Graph theory and applications16 references3 citations

TL;DR

This paper computes the energy of commuting graphs for various finite non-Abelian AC-groups—groups where all centralizers of non-central elements are Abelian—using spectral graph theory. It derives explicit formulas for energy based on group structure, particularly the size of the center and quotient groups, and proves that these commuting graphs are neither hyperenergetic nor borderenergetic, offering a broad generalization of energy bounds for such graphs.

ABSTRACT

Let $Γ$ be a graph with the adjacency matrix $A$. The energy of $Γ$ is the sum of the absolute values of the eigenvalues of $A$. In this article we compute the energies of the commuting graphs of some finite groups and discuss some consequences.

Motivation & Objective

• To compute the energy of commuting graphs for specific families of finite non-Abelian AC-groups.
• To establish explicit formulas for energy based on group invariants such as order, center size, and quotient structure.
• To investigate whether these commuting graphs are hyperenergetic or borderenergetic, based on comparison with complete graphs.
• To generalize energy bounds across classes of groups with Abelian centralizers, particularly those with quotients isomorphic to Z_p×Z_p or D_2m.
• To pose a broader conjecture on the non-hyperenergetic nature of commuting graphs across all finite non-Abelian groups.

Proposed method

• Uses spectral graph theory: energy is defined as the sum of absolute values of eigenvalues of the adjacency matrix.
• Applies known spectral results from prior works (e.g., [6, 7]) to compute eigenvalues and multiplicities for commuting graphs.
• Leverages the structure of AC-groups and the fact that centralizers are Abelian to decompose the commuting graph into disjoint unions of complete graphs.
• Employs the identity that the number of vertices in Γ_G is |G| − |Z(G)|, and uses Lemma 1 to relate centralizer sizes to group order.
• Derives energy formulas via the general formula E(Γ_G) = 2(|G| − |Z(G)| − n), where n is the number of distinct non-central centralizers.
• Compares computed energy values with the energy of the complete graph on the same number of vertices to assess hyperenergetic or borderenergetic status.

Experimental results

Research questions

• RQ1 What is the exact energy of the commuting graph for the group M_{2mn} when m is odd or even?
• RQ2 How does the energy of the commuting graph of D_{2m} depend on the parity of m?
• RQ3 Is the commuting graph of Q_{4m} hyperenergetic or borderenergetic, and what is its energy?
• RQ4 Under what conditions is the commuting graph of a finite non-Abelian group neither hyperenergetic nor borderenergetic?
• RQ5 Does the commuting graph of every finite non-Abelian AC-group fail to be hyperenergetic or borderenergetic?

Key findings

• For M_{2mn}, the energy is 4mn − 2m − 2n − 2 if m is odd, and 4mn − 4n − m − 2 if m is even.
• For D_{2m}, the energy is 2m − 4 if m is odd, and 3m − 6 if m is even.
• For Q_{4m}, the energy is exactly 6m − 3, and this graph is neither hyperenergetic nor borderenergetic.
• For U_{6n}, the energy is 10n − 8, and the graph is neither hyperenergetic nor borderenergetic.
• For QD_{2^n}, the energy is 2^n + 2^{n−1} − 6, and the graph is neither hyperenergetic nor borderenergetic.
• For groups with G/Z(G) ≅ Z_p × Z_p or D_{2m}, the energy is 2((p² − 1)|Z(G)| − p − 1) or 2((2m − 1)|Z(G)| − m − 1), respectively, and such graphs are neither hyperenergetic nor borderenergetic.