The second largest eigenvalue and vertex-connectivity of regular multigraphs

TL;DR

This paper extends Fiedler's inequality relating algebraic connectivity and vertex connectivity from simple graphs to multigraphs, establishing bounds involving maximum edge multiplicity and constructing extremal examples.

Contribution

It generalizes the inequality between algebraic connectivity and vertex connectivity to multigraphs and provides sharp bounds and constructions for regular multigraphs.

Findings

Extended Fiedler's inequality to multigraphs involving maximum edge multiplicity.

Proved that for regular multigraphs, algebraic connectivity bounds vertex connectivity.

Constructed examples showing the bounds are tight.

Abstract

Let $\mu_2(G)$ be the second smallest Laplacian eigenvalue of a graph $G$. The vertex connectivity of $G$, written $\kappa(G)$, is the minimum size of a vertex set $S$ such that $G-S$ is disconnected. Fiedler proved that $\mu_2(G) \le \kappa(G)$ for a non-complete simple graph $G$; for this reason $\mu_2(G)$ is called the "algebraic connectivity" of $G$. We extend his result to multigraphs. For a pair of vertices $u$ and $v$, let $m(u,v)$ be the number of edges with endpoints $u$ and $v$. For a vertex $v$, let $m(v)=\max_{u \in N(v)} m(v,u)$, where $N(v)$ is the set of neighbors of $v$, and let $m(G)=\max_{v \in V(G)} m(v)$. We prove that for any multigraph $G$ whose underlying graph is not a complete graph, $\mu_2(G) \le \kappa(G) m(G)$. We also prove that for any $d$-regular multigraph $G$ whose underlying graph is not the complete graph with 2 vertices, if $\mu_2(G) > \frac d4$, then $G$ is 2-connected. For $t\ge2$ and infinitely many $d$, we construct $d$-regular multigraphs $H$ with $\mu_2(H)=d$, $\kappa(H)=t$, and $m(H)=\frac dt$. These graphs show that the inequality $\mu_2(G) \le \kappa(G) m(G)$ is sharp. In addition, we prove that if $G$ is a $d$-regular multigraph whose underlying graph is not a complete graph, then $\mu_2(G) \le d$; equality holds for the graphs in the construction.