Connectivity, toughness, spanning trees of bounded degree, and the spectrum of regular graphs

TL;DR

This paper explores the relationships between the spectral properties of regular graphs and their structural features such as connectivity, toughness, and spanning trees with bounded degree, providing new theoretical insights.

Contribution

It introduces novel results linking the spectrum of regular graphs to their connectivity, toughness, and spanning tree properties, expanding understanding of graph spectral theory.

Findings

Spectral conditions imply bounds on graph connectivity.

New criteria for toughness based on eigenvalues.

Existence of bounded degree spanning trees related to spectral properties.

Abstract

In this paper, we present some new results describing connections between the spectrum of a regular graph and its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.