A spectral condition for spanning trees with restricted degrees in bipartite graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917357130088448 |
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| author | Wu, Jiancheng Zhou, Sizhong Liu, Hongxia |
| author_facet | Wu, Jiancheng Zhou, Sizhong Liu, Hongxia |
| contents | Let $G$ be a graph and $T$ be a spanning tree of $G$. We use $Q(G)=D(G)+A(G)$ to denote the signless Laplacian matrix of $G$, where $D(G)$ is the diagonal degree matrix of $G$ and $A(G)$ is the adjacency matrix of $G$. The signless Laplacian spectral radius of $G$ is denoted by $q(G)$. A necessary and sufficient condition for a connected bipartite graph $G$ with bipartition $(A,B)$ to have a spanning tree $T$ with $d_T(v)\geq k$ for any $v\in A$ was independently obtained by Frank and Gyárfás (A. Frank, E. Gyárfás, How to orient the edges of a graph?, Colloq. Math. Soc. Janos Bolyai 18 (1976) 353--364), Kaneko and Yoshimoto (A. Kaneko, K. Yoshimoto, On spanning trees with restricted degrees, Inform. Process. Lett. 73 (2000) 163--165). Based on the above result, we establish a lower bound on the signless Laplacian spectral radius $q(G)$ of a connected bipartite graph $G$ with bipartition $(A,B)$, in which the bound guarantees that $G$ has a spanning tree $T$ with $d_T(v)\geq k$ for any $v\in A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00700 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A spectral condition for spanning trees with restricted degrees in bipartite graphs Wu, Jiancheng Zhou, Sizhong Liu, Hongxia Combinatorics 05C50, 05C05, 05C70 Let $G$ be a graph and $T$ be a spanning tree of $G$. We use $Q(G)=D(G)+A(G)$ to denote the signless Laplacian matrix of $G$, where $D(G)$ is the diagonal degree matrix of $G$ and $A(G)$ is the adjacency matrix of $G$. The signless Laplacian spectral radius of $G$ is denoted by $q(G)$. A necessary and sufficient condition for a connected bipartite graph $G$ with bipartition $(A,B)$ to have a spanning tree $T$ with $d_T(v)\geq k$ for any $v\in A$ was independently obtained by Frank and Gyárfás (A. Frank, E. Gyárfás, How to orient the edges of a graph?, Colloq. Math. Soc. Janos Bolyai 18 (1976) 353--364), Kaneko and Yoshimoto (A. Kaneko, K. Yoshimoto, On spanning trees with restricted degrees, Inform. Process. Lett. 73 (2000) 163--165). Based on the above result, we establish a lower bound on the signless Laplacian spectral radius $q(G)$ of a connected bipartite graph $G$ with bipartition $(A,B)$, in which the bound guarantees that $G$ has a spanning tree $T$ with $d_T(v)\geq k$ for any $v\in A$. |
| title | A spectral condition for spanning trees with restricted degrees in bipartite graphs |
| topic | Combinatorics 05C50, 05C05, 05C70 |
| url | https://arxiv.org/abs/2412.00700 |