Some results on Hamming graphs and an extended Hamming graphs

TL;DR

This paper introduces the extended Hamming graph, explores its algebraic and structural properties, and determines its spectrum, revealing new insights into its diameter, Cayley graph status, and eigenvalues.

Contribution

It presents the novel extended Hamming graph, analyzes its properties, and derives its spectrum, expanding understanding of graph spectra and structural characteristics.

Findings

Diameter of EH(n,2^n) is n

EH(n,2^n) is a Cayley graph but not distance regular

Spectrum of EH(n,2^n) is λ_i ± 1, with explicit eigenvalue multiplicities

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 $\lambda_i\pm 1$, where $\lambda_i$ are the eigenvalues of the underlying Hamming graph $H(n,2^n)$. The multiplicity of each eigenvalue is explicitly calculated.