Some results on Hamming graphs and an extended Hamming graphs
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911325772316672 |
|---|---|
| author | Zafari, Ali Alikhani, Saeid |
| author_facet | Zafari, Ali Alikhani, Saeid |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16490 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some results on Hamming graphs and an extended Hamming graphs Zafari, Ali Alikhani, Saeid Combinatorics 05C25, 94C15 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. |
| title | Some results on Hamming graphs and an extended Hamming graphs |
| topic | Combinatorics 05C25, 94C15 |
| url | https://arxiv.org/abs/2512.16490 |