Spanning subgraphs and spectral radius in graphs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918092694618112 |
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| author | Zhou, Sizhong |
| author_facet | Zhou, Sizhong |
| contents | A spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ that is a tree covers all vertices of $G$. The leaf distance of $T$ is defined as the minimum of distances between any two leaves of $T$. A fractional matching of a graph $G$ is a function $h$ assigning every edge a real number in $[0,1]$ so that $\sum\limits_{e\in E_G(v)}{h(e)}\leq1$ for any $v\in V(G)$, where $E_G(v)$ denotes the set of edges incident with $v$ in $G$. A fractional matching of $G$ is called a fractional perfect matching if $\sum\limits_{e\in E_G(v)}{h(e)}=1$ for any $v\in V(G)$. A graph $G$ with at least $2k+2$ vertices is said to be fractional $k$-extendable if every $k$-matching $M$ in $G$ is included in a fractional perfect matching $h$ of $G$ such that $h(e)=1$ for any $e\in M$. This paper considers a lower bound on the spectral radius of $G$ to guarantee that $G$ has a spanning tree with leaf distance at least $d$. At the same time, we obtain a lower bound on the spectral radius of $G$ to ensure that $G$ is fractional $k$-extendable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11078 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spanning subgraphs and spectral radius in graphs Zhou, Sizhong Combinatorics 05C50, 05C05, 05C70 A spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ that is a tree covers all vertices of $G$. The leaf distance of $T$ is defined as the minimum of distances between any two leaves of $T$. A fractional matching of a graph $G$ is a function $h$ assigning every edge a real number in $[0,1]$ so that $\sum\limits_{e\in E_G(v)}{h(e)}\leq1$ for any $v\in V(G)$, where $E_G(v)$ denotes the set of edges incident with $v$ in $G$. A fractional matching of $G$ is called a fractional perfect matching if $\sum\limits_{e\in E_G(v)}{h(e)}=1$ for any $v\in V(G)$. A graph $G$ with at least $2k+2$ vertices is said to be fractional $k$-extendable if every $k$-matching $M$ in $G$ is included in a fractional perfect matching $h$ of $G$ such that $h(e)=1$ for any $e\in M$. This paper considers a lower bound on the spectral radius of $G$ to guarantee that $G$ has a spanning tree with leaf distance at least $d$. At the same time, we obtain a lower bound on the spectral radius of $G$ to ensure that $G$ is fractional $k$-extendable. |
| title | Spanning subgraphs and spectral radius in graphs |
| topic | Combinatorics 05C50, 05C05, 05C70 |
| url | https://arxiv.org/abs/2507.11078 |