Some results on Hamming graphs and an extended Hamming graphs
Ali Zafari, Saeid Alikhani
TL;DR
This work investigates generalized Hamming graphs by introducing the extended Hamming graph $EH(n,2^n)$, derived from $H(n,2^n)$ with added complementary edges. It first rederives the spectrum of the folded hypercube, linking folded and hypercube eigenvalues via $\theta_i = \lambda_i + (-1)^i$. For $EH(n,2^n)$, the authors establish a diameter of $n$, prove Cayley graph structure on the abelian group $(\mathbb{F}_{2^n})^n$, show non-distance-regularity, and derive the complete spectrum with explicit multiplicities given by $\theta_{i,t} = \lambda_i + (-1)^t$. These results deepen the understanding of highly symmetric generalized Hamming graphs and provide a framework for further spectral and combinatorial investigations of EH-type graphs.
Abstract
In this paper we first obtain the spectrum of the folded hypercube in a new approach. Then we introduce a new family of graphs called the extended Hamming graph, denoted by $EH(n,2^n)$, which is constructed from the well-known Hamming graph $H(n,2^n)$. The graph $EH(n,2^n)$ shares the same vertex set as $H(n,2^n)$ but includes additional edges, called complementary edges, connecting each $n$-tuple vertex $u$ to its complement $u^c$, where $u^c$ is defined such that the sum of each two corresponding coordinates of $u$ and $u^c$ equals $2^n-1$. We investigate several algebraic and structural properties of this new family of graphs. Specifically, we show that the diameter of $EH(n,2^n)$ is $n$. We prove that $EH(n,2^n)$ is a Cayley graph, but we demonstrate that it is not a distance regular graph. Finally, we determine the spectrum of $EH(n,2^n)$, showing that its eigenvalues are $λ_i\pm 1$, where $λ_i$ are the eigenvalues of the underlying Hamming graph $H(n,2^n)$. The multiplicity of each eigenvalue is explicitly calculated.