Equitable-partition methods for the hamming spectrum and energy of graph families

Abstract

The Hamming matrix H(G) of a graph G encodes the Hamming distances between the binary incidence strings of its vertices and gives rise to Hamming spectrum and Hamming energy HE(G). Building on recent foundational paper that established bounds for paths and closed-form formulas for regular graphs, we develop a unified approach via equitable partitions to determine the full Hamming spectrum and energy of several graph families, such as semi-regular bipartite graphs, complete multipartite graphs, wheel graphs, windmill graphs and the Dutch windmill graphs. In each of these graph classes we partition the vertex sets into a small number of natural classes so that the corresponding quotient matrix B is only \(2\times 2\) (or \(3\times 3\) in the Dutch windmill case). Diagonalizing these low-dimensional matrices yields explicit eigenvalues for H(G) and hence closed-form expressions for HE(G). Our results significantly extend the list of graphs for which Hamming-based spectral invariants are known in closed form, and illustrate the power of equitable partitions in spectral graph theory.