Spanning k-trees, odd [1,b]-factors and spectral radius in binding graphs
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866912473631686656 |
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| author | Wu, Jiancheng Zhou, Sizhong |
| author_facet | Wu, Jiancheng Zhou, Sizhong |
| contents | The binding number of a graph $G$, written as $\mbox{bind}(G)$, is defined by $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}. $$ A graph $G$ is called $r$-binding if $\mbox{bind}(G)\geq r$. An odd $[1,b]$-factor of a graph $G$ is a spanning subgraph $F$ with $d_F(v)\in\{1,3,\ldots,b\}$ for all $v\in V(G)$, where $b\geq1$ is an odd integer. A spanning $k$-tree of a connected graph $G$ is a spanning tree $T$ with $d_T(v)\leq k$ for every $v\in V(G)$. In this paper, we first show a tight sufficient condition with respect to the adjacency spectral radius for connected $\frac{1}{b}$-binding graphs to have odd $[1,b]$-factors, which generalizes Fan and Lin's previous result [D. Fan, H. Lin, Binding number, $k$-factor and spectral radius of graphs, Electron. J. Combin. 31(1) (2024) \#P1.30] and partly improves Fan, Liu and Ao's previous result [A. Fan, R. Liu, G. Ao, Spectral radius, odd $[1,b]$-factor and spanning $k$-tree of 1-binding graphs, Linear Algebra Appl. 705 (2025) 1--16]. Then we put forward a tight sufficient condition via the adjacency spectral radius for connected $\frac{1}{k-2}$-binding graphs to have spanning $k$-trees, which partly improves Fan, Liu and Ao's previous result [A. Fan, R. Liu, G. Ao, Spectral radius, odd $[1,b]$-factor and spanning $k$-tree of 1-binding graphs, Linear Algebra Appl. 705 (2025) 1--16]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_07301 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spanning k-trees, odd [1,b]-factors and spectral radius in binding graphs Wu, Jiancheng Zhou, Sizhong Combinatorics 05C50, 05C05, 05C70 The binding number of a graph $G$, written as $\mbox{bind}(G)$, is defined by $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}. $$ A graph $G$ is called $r$-binding if $\mbox{bind}(G)\geq r$. An odd $[1,b]$-factor of a graph $G$ is a spanning subgraph $F$ with $d_F(v)\in\{1,3,\ldots,b\}$ for all $v\in V(G)$, where $b\geq1$ is an odd integer. A spanning $k$-tree of a connected graph $G$ is a spanning tree $T$ with $d_T(v)\leq k$ for every $v\in V(G)$. In this paper, we first show a tight sufficient condition with respect to the adjacency spectral radius for connected $\frac{1}{b}$-binding graphs to have odd $[1,b]$-factors, which generalizes Fan and Lin's previous result [D. Fan, H. Lin, Binding number, $k$-factor and spectral radius of graphs, Electron. J. Combin. 31(1) (2024) \#P1.30] and partly improves Fan, Liu and Ao's previous result [A. Fan, R. Liu, G. Ao, Spectral radius, odd $[1,b]$-factor and spanning $k$-tree of 1-binding graphs, Linear Algebra Appl. 705 (2025) 1--16]. Then we put forward a tight sufficient condition via the adjacency spectral radius for connected $\frac{1}{k-2}$-binding graphs to have spanning $k$-trees, which partly improves Fan, Liu and Ao's previous result [A. Fan, R. Liu, G. Ao, Spectral radius, odd $[1,b]$-factor and spanning $k$-tree of 1-binding graphs, Linear Algebra Appl. 705 (2025) 1--16]. |
| title | Spanning k-trees, odd [1,b]-factors and spectral radius in binding graphs |
| topic | Combinatorics 05C50, 05C05, 05C70 |
| url | https://arxiv.org/abs/2507.07301 |