New conjectures on algebraic connectivity and the Laplacian spread of graphs

TL;DR

This paper proposes new conjectures relating algebraic connectivity and Laplacian spread in graphs, including bounds involving high eccentricity vertices, and discusses their implications and extensions for simple and weighted graphs.

Contribution

It introduces new conjectures on algebraic connectivity and Laplacian spread, providing theoretical bounds and strengthening existing conjectures in graph theory.

Findings

Proposed a new lower bound on algebraic connectivity involving high eccentricity vertices.

Showed that this bound implies a stronger Laplacian Spread Conjecture.

Discussed extensions to simple and weighted graphs.

Abstract

We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for simple graphs and a conjecture for weighted graphs.