Some results on Hamming graphs and an extended Hamming graphs

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.