Spanning k-trees, odd [1,b]-factors and spectral radius in binding graphs

The binding number of a graph $G$, written as $bind(G)$, is defined by $$bind(G)=\min \{\frac{\mathrm{\mid }{N}_G(X)\mathrm{\mid }}{\mathrm{\mid }X\mathrm{\mid }}:\mathrm{\varnothing }\mathrm{\ne }X\subseteq V(G),N_G(X)\mathrm{\ne }V(G)\}\mathrm{.}$$ A graph $G$ is called $r$-binding if $bind(G)\ge r$. An odd $[1,b]$-factor of a graph $G$ is a spanning subgraph $F$ with $d_F(v)\in \{1,3,\dots ,b\}$ for all $v\in V(G)$, where $b\ge 1$ is an odd integer. A spanning $k$-tree of a connected graph $G$ is a spanning tree $T$ with $d_T(v)\le k$ 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].