Eigenvalues and spanning trees with constrained degree
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2023
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| author | Liu, Chang Li, Jianping |
| author_facet | Liu, Chang Li, Jianping |
| contents | In this paper, we study some spanning trees with bounded degree and leaf degree from eigenvalues. For any integer $k\geq2$, a $k$-tree is a spanning tree in which every vertex has degree no more than $k$. Let $T$ be a spanning tree of a connected graph. The leaf degree of $T$ is the maximum number of end-vertices attached to $v$ in $T$ for any $v\in V(T)$. By referring to the technique shown in [Eigenvalues and $[a,b]$-factors in regular graphs, J. Graph Theory. 100 (2022) 458-469], for an $r$-regular graph $G$, we provide an upper bound for the fourth largest adjacency eigenvalue of $G$ to guarantee the existence of a $k$-tree. Moreover, for a $t$-connected graph, we prove a tight sufficient condition for the existence of a spanning tree with leaf degree at most $k$ in terms of spectral radius. This generalizes a result of Theorem 1.5 in [Spectral radius and spanning trees of graphs, Discrete Math. 346 (2023) 113400]. Finally, for a general graph $G$, we present two sufficient conditions for the existence of a spanning tree with leaf degree at most $k$ via the Laplacian eigenvalues of $G$ and the spectral radius of the complement of $G$, respectively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_12417 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Eigenvalues and spanning trees with constrained degree Liu, Chang Li, Jianping Combinatorics 05C50, 05C70 In this paper, we study some spanning trees with bounded degree and leaf degree from eigenvalues. For any integer $k\geq2$, a $k$-tree is a spanning tree in which every vertex has degree no more than $k$. Let $T$ be a spanning tree of a connected graph. The leaf degree of $T$ is the maximum number of end-vertices attached to $v$ in $T$ for any $v\in V(T)$. By referring to the technique shown in [Eigenvalues and $[a,b]$-factors in regular graphs, J. Graph Theory. 100 (2022) 458-469], for an $r$-regular graph $G$, we provide an upper bound for the fourth largest adjacency eigenvalue of $G$ to guarantee the existence of a $k$-tree. Moreover, for a $t$-connected graph, we prove a tight sufficient condition for the existence of a spanning tree with leaf degree at most $k$ in terms of spectral radius. This generalizes a result of Theorem 1.5 in [Spectral radius and spanning trees of graphs, Discrete Math. 346 (2023) 113400]. Finally, for a general graph $G$, we present two sufficient conditions for the existence of a spanning tree with leaf degree at most $k$ via the Laplacian eigenvalues of $G$ and the spectral radius of the complement of $G$, respectively. |
| title | Eigenvalues and spanning trees with constrained degree |
| topic | Combinatorics 05C50, 05C70 |
| url | https://arxiv.org/abs/2311.12417 |