Toughness and spectral radius in graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914832929783808 |
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| author | Wang, Sufang Zhang, Wei |
| author_facet | Wang, Sufang Zhang, Wei |
| contents | Let $t$ be a positive integer, and let $G$ be a connected graph of order $n$ with $n\geq t+2$. A graph $G$ is said to be $\frac{1}{t}$-tough if $|S|\geq\frac{1}{t}c(G-S)$ for every subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is the number of connected components in $G-S$. The adjacency matrix of $G$ is denoted by $A(G)$. Let $λ_1(G)\geqλ_2(G)\geq\dots\geqλ_n(G)$ be the eigenvalues of $A(G)$. In particular, the eigenvalue $λ_1(G)$ is called the spectral radius of $G$. In this paper, we prove that $G$ is a $\frac{1}{t}$-tough graph unless $G=K_1\vee(K_{n-t-1}\cup tK_1)$ if $λ_1(G)\geqη(t,n)$, where $η(t,n)$ is the largest root of $x^{3}-(n-t-2)x^{2}-(n-1)x+t(n-t-2)=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08224 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toughness and spectral radius in graphs Wang, Sufang Zhang, Wei Combinatorics 05C50, 90B99 Let $t$ be a positive integer, and let $G$ be a connected graph of order $n$ with $n\geq t+2$. A graph $G$ is said to be $\frac{1}{t}$-tough if $|S|\geq\frac{1}{t}c(G-S)$ for every subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is the number of connected components in $G-S$. The adjacency matrix of $G$ is denoted by $A(G)$. Let $λ_1(G)\geqλ_2(G)\geq\dots\geqλ_n(G)$ be the eigenvalues of $A(G)$. In particular, the eigenvalue $λ_1(G)$ is called the spectral radius of $G$. In this paper, we prove that $G$ is a $\frac{1}{t}$-tough graph unless $G=K_1\vee(K_{n-t-1}\cup tK_1)$ if $λ_1(G)\geqη(t,n)$, where $η(t,n)$ is the largest root of $x^{3}-(n-t-2)x^{2}-(n-1)x+t(n-t-2)=0$. |
| title | Toughness and spectral radius in graphs |
| topic | Combinatorics 05C50, 90B99 |
| url | https://arxiv.org/abs/2406.08224 |