Eccentricity and algebraic connectivity of graphs

TL;DR

This paper establishes new bounds relating the algebraic connectivity of a graph to its eccentricity distribution and diameter, providing insights into the graph's structural properties.

Contribution

It introduces novel inequalities linking algebraic connectivity with eccentricity and graph powers, advancing understanding of graph spectral and structural relationships.

Findings

Lower bounds on algebraic connectivity based on eccentricity counts

Bounds involving graph diameter and algebraic connectivity

Inequalities connecting algebraic connectivity with graph powers

Abstract

Let $G$ be a graph on $n$ nodes with algebraic connectivity $\lambda_{2}$. The eccentricity of a node is defined as the length of a longest shortest path starting at that node. If $s_\ell$ denotes the number of nodes of eccentricity at most $\ell$, then for $\ell \ge 2$, $$\lambda_{2} \ge \frac{ 4 \, s_\ell }{ (\ell-2+\frac{4}{n}) \, n^2 }.$$ As a corollary, if $d$ denotes the diameter of $G$, then $$\lambda_{2} \ge \frac{ 4 }{ (d-2+\frac{4}{n}) \, n }.$$ It is also shown that $$\lambda_{2} \ge \frac{ s_\ell }{ 1+ \ell \left(e(G^{\ell})-m\right) },$$ where $m$ and $e(G^\ell)$ denote the number of edges in $G$ and in the $\ell$-th power of $ G $, respectively.