Abstract

For a simple graph $G$, the Hamming matrix $H(G)$ is defined in terms of vertex degrees, and its eigenvalues constitute the Hamming spectrum of $G$. The sum of the absolute values of these eigenvalues is called the Hamming energy, denoted by $HE(G)$. In this paper, using equitable partitions, the Hamming spectrum and Hamming energy are explicitly determined for several classes of graphs, including double-wheel graphs, Dutch windmill graphs $D_{5}^{m}$, bistar graphs, and the corona product $K_{3}\circ\overline{K}_{m}$. In each family, the vertex set is partitioned so that the quotient matrix has small dimension (at most $4\times4$), which enables direct computation of eigenvalues. The results contribute new closed forms of $HE(G)$ and extend the collection of graphs whose complete Hamming spectrum is known.

Keywords: Hamming energy; double-wheel; Dutch windmill; corona product; bistar graph.

MSC: 05C09, 05C50

DOI: 10.57016/MV-MnZO6936

Pages:  1--15