Spectral radius and spanning trees of graphs

TL;DR

This paper establishes precise spectral conditions that ensure the existence of spanning trees with limited leaves or leaf degrees in connected graphs, extending previous theorems and characterizing extremal cases.

Contribution

It provides tight spectral criteria for spanning trees with at most k leaves or leaf degree constraints, generalizing and strengthening existing results.

Findings

Derived spectral conditions guaranteeing spanning k-ended-trees.

Characterized extremal graphs for these spectral conditions.

Extended previous theorems to broader classes of spanning trees.

Abstract

For integer $k\geq2,$ a spanning $k$-ended-tree is a spanning tree with at most $k$ leaves. Motivated by the closure theorem of Broersma and Tuinstra [Independence trees and Hamilton cycles, J. Graph Theory 29 (1998) 227--237], we provide tight spectral conditions to guarantee the existence of a spanning $k$-ended-tree in a connected graph of order $n$ with extremal graphs being characterized. Moreover, by adopting Kaneko's theorem [Spanning trees with constraints on the leaf degree, Discrete Appl. Math. 115 (2001) 73--76], we also present tight spectral conditions for the existence of a spanning tree with leaf degree at most $k$ in a connected graph of order $n$ with extremal graphs being determined, where $k\geq1$ is an integer.