Distance spectral radius and perfect matchings in graphs with given fractional property
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| Format: | Preprint |
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2026
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| _version_ | 1866915921291902976 |
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| author | Zhou, Sizhong |
| author_facet | Zhou, Sizhong |
| contents | A matching in a graph $G$ is a set of independent edges in $G$. A perfect matching in a graph $G$ is a matching which saturates all the vertices of $G$. A fractional perfect matching in a graph $G$ is a function $h:E(G)\rightarrow [0,1]$ such that $\sum\limits_{e\in E_G(v)}h(e)=1$ for every $v\in V(G)$, where $E_G(v)$ is the set of edges incident to $v$ in $G$. Clearly, the existence of a fractional perfect matching in a graph is a necessary condition for the graph to possess a perfect matching. Let $G$ be a $k$-connected graph of even order $n$ with a fractional perfect matching, where $k$ is a positive integer. We denote by $μ(G)$ the distance spectral radius of $G$. In this paper, we prove that if $n\geq8k+6$ and $μ(G)\leqμ(K_k\vee(kK_1\cup K_3\cup K_{n-2k-3}))$, then $G$ contains a perfect matching unless $G=K_k\vee(kK_1\cup K_3\cup K_{n-2k-3})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_05869 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Distance spectral radius and perfect matchings in graphs with given fractional property Zhou, Sizhong Combinatorics 05C50, 05C70 A matching in a graph $G$ is a set of independent edges in $G$. A perfect matching in a graph $G$ is a matching which saturates all the vertices of $G$. A fractional perfect matching in a graph $G$ is a function $h:E(G)\rightarrow [0,1]$ such that $\sum\limits_{e\in E_G(v)}h(e)=1$ for every $v\in V(G)$, where $E_G(v)$ is the set of edges incident to $v$ in $G$. Clearly, the existence of a fractional perfect matching in a graph is a necessary condition for the graph to possess a perfect matching. Let $G$ be a $k$-connected graph of even order $n$ with a fractional perfect matching, where $k$ is a positive integer. We denote by $μ(G)$ the distance spectral radius of $G$. In this paper, we prove that if $n\geq8k+6$ and $μ(G)\leqμ(K_k\vee(kK_1\cup K_3\cup K_{n-2k-3}))$, then $G$ contains a perfect matching unless $G=K_k\vee(kK_1\cup K_3\cup K_{n-2k-3})$. |
| title | Distance spectral radius and perfect matchings in graphs with given fractional property |
| topic | Combinatorics 05C50, 05C70 |
| url | https://arxiv.org/abs/2604.05869 |