The spanning $k$-trees, perfect matchings and spectral radius of graphs

TL;DR

This paper establishes spectral radius conditions that guarantee the existence of $k$-trees and perfect matchings in certain classes of graphs, linking spectral properties to combinatorial structures.

Contribution

It introduces new spectral radius criteria for the existence of $k$-trees and perfect matchings in graphs, expanding the understanding of spectral graph theory.

Findings

Spectral radius bounds ensure $k$-trees in connected graphs.

Spectral conditions guarantee perfect matchings in bipartite graphs.

Provides new spectral criteria linking graph spectra to combinatorial properties.

Abstract

A $k$-tree is a spanning tree in which every vertex has degree at most $k$. In this paper, we provide a sufficient condition for the existence of a $k$-tree in a connected graph with fixed order in terms of the adjacency spectral radius and the signless Laplacian spectral radius, respectively. Also, we give a similar condition for the existence of a perfect matching in a balanced bipartite graph with fixed order and minimum degree.