Common Neighborhood Energy of the Non-Commuting Graphs and Commuting Graphs Associated with Dihedral and Generalized Quaternion Groups

This paper explores the common neighborhood energy ( E C N ( Γ ) ) of graphs derived from the dihedral group D 2 n and generalized quaternion group Q 4 n , specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). Studying graphs associated with groups offers a powerful approach to translating algebraic properties into combinatorial structures, enabling the application of graph-theoretic tools to understand group behavior. The common neighborhood energy, defined as the sum of the absolute values of the eigenvalues of the common neighborhood (CN) matrix, i.e., ∑ i = 1 p | ζ i | , where { ζ i } i = 1 p are the CN eigenvalues, provides insights into the structural properties of these graphs. We derive explicit formulas for the CN characteristic polynomials and corresponding CN eigenvalues for both the NCM-graph and CM-graph as functions of n . Consequently, we establish closed-form expressions for the E C N of these graphs, which are parameterized by n . The validity of our theoretical results is confirmed through computational examples. This study contributes to the spectral analysis of algebraic graphs, demonstrating a direct connection between the group-theoretic structure of D 2 n and Q 4 n , as well as the combinatorial energy of their associated graphs, thus furthering the understanding of group properties through spectral graph theory.