Hamming energy of certain graph products derived from regular circulant graphs

Abstract

Let $G$ be a graph of order $n$. The hamming matrix $H(G) = [h_{ij}]$ of $G$ is an $n \times n$ matrix whose $(i,j)$-entry is the hamming distance between the strings $s(v_i)$ and $s(v_j)$. The hamming energy $HE(G)$ of a graph $G$ is the sum of the absolute values of the eigenvalues of $H(G)$. In this paper, we study the hamming energy of the tensor product \( G_1 \otimes G_2 \), the lexicographic product \( G_1[G_2] \) of regular circulant graphs by establishing the relationship between the spectrum of hamming matrix of these graphs with the adjacency spectrum of $G_1$ and $G_2$. We also obtain spectrum of join of a finite collection of regular graphs in terms of spectrum of the component regular graphs.