Cayley graphs on symmetric groups generated by $n$-cycles are hyperenergetic
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912073306341376 |
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| author | Ebrahimi, Mahdi |
| author_facet | Ebrahimi, Mahdi |
| contents | Let $Γ$ be a simple graph with $n$ vertices. The energy of $Γ$, denoted by $\mathcal{E}(Γ)$, is defined as the sum of the absolute values of the eigenvalues of $Γ$. The graph $Γ$ is said to be hyperenergetic if $\mathcal{E}(Γ)>2n-2$. For the graph $Γ$, the multiplicity of the eigenvalue $0$, denoted by $η(Γ)$, is called the nullity of $Γ$. In this paper, we show that for every positive integer $n\geq 4$, the Cayley graph $Γ_n$ on the symmetric group $\mathrm{Sym}(n)$ generated by $n$-cycles is an integral hyperenergetic graph with $\mathcal{E}(Γ_n)=2^{n-1}(n-1)!$ and $η(Γ_n)=n!-\binom{2n-2}{n-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11306 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cayley graphs on symmetric groups generated by $n$-cycles are hyperenergetic Ebrahimi, Mahdi Combinatorics 05C92, 20C30, 05C50 Let $Γ$ be a simple graph with $n$ vertices. The energy of $Γ$, denoted by $\mathcal{E}(Γ)$, is defined as the sum of the absolute values of the eigenvalues of $Γ$. The graph $Γ$ is said to be hyperenergetic if $\mathcal{E}(Γ)>2n-2$. For the graph $Γ$, the multiplicity of the eigenvalue $0$, denoted by $η(Γ)$, is called the nullity of $Γ$. In this paper, we show that for every positive integer $n\geq 4$, the Cayley graph $Γ_n$ on the symmetric group $\mathrm{Sym}(n)$ generated by $n$-cycles is an integral hyperenergetic graph with $\mathcal{E}(Γ_n)=2^{n-1}(n-1)!$ and $η(Γ_n)=n!-\binom{2n-2}{n-1}$. |
| title | Cayley graphs on symmetric groups generated by $n$-cycles are hyperenergetic |
| topic | Combinatorics 05C92, 20C30, 05C50 |
| url | https://arxiv.org/abs/2410.11306 |