Hamming energy of sunlet and n-barbell graphs

Abstract

The Hamming matrix of a graph arises from the notion of Hamming distance and provides a matrix-based framework for studying vertex dissimilarity. The corresponding Hamming energy, defined as the sum of the absolute values of the eigenvalues of the Hamming matrix, represents a natural spectral invariant that is closely related to the classical graph energy. In this paper, we investigate the Hamming matrix of two important families of graphs, namely sunlet graphs and barbell graphs. By applying the technique of equitable vertex partitions and methods from matrix spectral theory, we obtain explicit expressions for the H-spectrum and the H-energy of these graphs. Our results extend and complement existing studies on energy-like invariants for special classes of graphs.