2025/07/09 by Wu, Jiancheng, Zhou, Sizhong
#05C05 #05C50 #05C70 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.07301
The binding number of a graph G, written as bind(G), is defined by bind(G)=min\(|NG(X)|)/(|X|):∅≠ X⊆ V(G),NG(X)≠ V(G)\. A graph G is called r-binding if bind(G)≥ r. An odd [1,b]-factor of a graph G is a spanning subgraph F with dF(v)∈\1,3,…,b\ for all v∈ V(G), where b≥1 is an odd integer. A spanning k-tree of a connected graph G is a spanning tree T with dT(v)≤ k for every v∈ V(G). In this paper, we first show a tight sufficient condition with respect to the adjacency spectral radius for connected (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 (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].