Maximizing algebraic connectivity for certain families of graphs

TL;DR

This paper explores bounds on the algebraic connectivity of various graph families, providing new upper bounds for trees and cubic graphs, and discusses related conjectures and open problems.

Contribution

It introduces novel upper bounds on algebraic connectivity for trees with degree constraints and cubic graphs with specified girth, improving existing bounds and proposing new conjectures.

Findings

Bound on algebraic connectivity for trees with degree d: 2(d-2)/n + O(ln n / n^2)

Upper bound on cubic graphs' algebraic connectivity involving girth g: 3 - 2^{3/2} cos(π/⌊g/2⌋)

Proposes conjectures and open questions on algebraic connectivity bounds

Abstract

We investigate the bounds on algebraic connectivity of graphs subject to constraints on the number of edges, vertices, and topology. We show that the algebraic connectivity for any tree on $n$ vertices and with maximum degree $d$ is bounded above by $2(d-2) \frac{1}{n}+O(\frac{\ln n}{n^{2}}) .$ We then investigate upper bounds on algebraic connectivity for cubic graphs. We show that algebraic connectivity of a cubic graph of girth $g$ is bounded above by $3-2^{3/2}\cos(\pi/\lfloor g/2\rfloor) ,$ which is an improvement over the bound found by Nilli [A. Nilli, Electron. J. Combin., 11(9), 2004]. Finally, we propose several conjectures and open questions.