A result on spanning trees with bounded total excess
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918402518417408 |
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| author | Zhou, Sizhong |
| author_facet | Zhou, Sizhong |
| contents | Let $G$ be a connected graph and $T$ a spanning tree of $G$. Let $ρ(G)$ denote the adjacency spectral radius of $G$. The $k$-excess of a vertex $v$ in $T$ is defined as $\max\{0,d_T(v)-k\}$. The total $k$-excess $\mbox{te}(T,k)$ is defined by $\mbox{te}(T,k)=\sum\limits_{v\in V(T)}{\max\{0,d_T(v)-k\}}$. A tree $T$ is said to be a $k$-tree if $d_T(v)\leq k$ for any $v\in V(T)$, that is to say, the maximum degree of a $k$-tree is at most $k$. In fact, $T$ is a spanning $k$-tree if and only if $\mbox{te}(T,k)=0$. This paper studies a generalization of spanning $k$-trees using a concept called total $k$-excess and proposes a lower bound for $ρ(G)$ in a connected graph $G$ to ensure that $G$ contains a spanning tree $T$ with $\mbox{te}(T,k)\leq b$, where $k$ and $b$ are two nonnegative integers with $k\geq\max\{5,b+3\}$ and $(b,k)\neq(2,5)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15139 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A result on spanning trees with bounded total excess Zhou, Sizhong Combinatorics 05C50, 05C05 Let $G$ be a connected graph and $T$ a spanning tree of $G$. Let $ρ(G)$ denote the adjacency spectral radius of $G$. The $k$-excess of a vertex $v$ in $T$ is defined as $\max\{0,d_T(v)-k\}$. The total $k$-excess $\mbox{te}(T,k)$ is defined by $\mbox{te}(T,k)=\sum\limits_{v\in V(T)}{\max\{0,d_T(v)-k\}}$. A tree $T$ is said to be a $k$-tree if $d_T(v)\leq k$ for any $v\in V(T)$, that is to say, the maximum degree of a $k$-tree is at most $k$. In fact, $T$ is a spanning $k$-tree if and only if $\mbox{te}(T,k)=0$. This paper studies a generalization of spanning $k$-trees using a concept called total $k$-excess and proposes a lower bound for $ρ(G)$ in a connected graph $G$ to ensure that $G$ contains a spanning tree $T$ with $\mbox{te}(T,k)\leq b$, where $k$ and $b$ are two nonnegative integers with $k\geq\max\{5,b+3\}$ and $(b,k)\neq(2,5)$. |
| title | A result on spanning trees with bounded total excess |
| topic | Combinatorics 05C50, 05C05 |
| url | https://arxiv.org/abs/2507.15139 |