Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2406.10497 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917812967047168 |
|---|---|
| author | Ebrahimi, Mahdi |
| author_facet | Ebrahimi, Mahdi |
| contents | For a graph $Γ$, the multiplicity of the eigenvalue $0$, denoted by $η(Γ)$, is called the nullity of $Γ$.
Also the energy of $Γ$, denoted by $\mathcal{E}(Γ)$, is defined as the sum of the absolute values of the eigenvalues of $Γ$. The index of a subgroup $H$ in a group $G$ is denoted by $[G:H]$.
For a prime $p$, let $G$ be a finite $p$-solvable group whose order is divisible by $p$. Also let $Ω_p(G)$ be the set of all $p$-singular elements of $G$.
In this paper, we apply block theory of finite groups to show that the Cayley graph $Γ_p(G):=\mathrm{Cay}(G,Ω_p(G))$ is an integral graph with $η(Γ_p(G))=|G|-[G:O_{p^\prime}(G)]$, where $O_{p^\prime}(G)$ is the largest normal subgroup of $G$ whose order is co-prime to $p$.
We also find a lower bound for $\mathcal{E}(Γ_p(G))$.
Finally, we prove that the diameter of $Γ_p(G)$ is at most $ |G|_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10497 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cayley graphs on $p$-solvable groups generated by $p$-singular elements Ebrahimi, Mahdi Combinatorics 05C50, 20C15, 20C20, 05C92 For a graph $Γ$, the multiplicity of the eigenvalue $0$, denoted by $η(Γ)$, is called the nullity of $Γ$. Also the energy of $Γ$, denoted by $\mathcal{E}(Γ)$, is defined as the sum of the absolute values of the eigenvalues of $Γ$. The index of a subgroup $H$ in a group $G$ is denoted by $[G:H]$. For a prime $p$, let $G$ be a finite $p$-solvable group whose order is divisible by $p$. Also let $Ω_p(G)$ be the set of all $p$-singular elements of $G$. In this paper, we apply block theory of finite groups to show that the Cayley graph $Γ_p(G):=\mathrm{Cay}(G,Ω_p(G))$ is an integral graph with $η(Γ_p(G))=|G|-[G:O_{p^\prime}(G)]$, where $O_{p^\prime}(G)$ is the largest normal subgroup of $G$ whose order is co-prime to $p$. We also find a lower bound for $\mathcal{E}(Γ_p(G))$. Finally, we prove that the diameter of $Γ_p(G)$ is at most $ |G|_p$. |
| title | Cayley graphs on $p$-solvable groups generated by $p$-singular elements |
| topic | Combinatorics 05C50, 20C15, 20C20, 05C92 |
| url | https://arxiv.org/abs/2406.10497 |