[Paper Review] Algebraic connectivity of multigraphs

O Suil|arXiv (Cornell University)|Mar 12, 2016

Graph theory and applications7 references3 citations

TL;DR

This paper extends Fiedler's result on algebraic connectivity to multigraphs by proving that for most multigraphs, the second smallest Laplacian eigenvalue μ₂(G) is bounded above by κ(G)·m(G), where κ(G) is the vertex connectivity and m(G) is the maximum edge multiplicity per vertex. The bound is sharp, as demonstrated by explicit constructions of d-regular multigraphs achieving equality.

ABSTRACT

Let $\mu_2(G)$ be the second smallest Laplacian eigenvalue of a graph $G$, and let $\kappa(G)$ be 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 $G$, except when $G$ is a multigraph obtained from a complete graph by duplicating edges, $\mu_2(G) \le \kappa(G) m(G)$. We also prove that for a $d$-regular multigraph $G$, except for the 2-vertex $d$-regular multigraph, 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, except for a graph obtained from a complete graph by duplicating edges, then $\mu_2(G) \le d$, equality holds for the graphs in the construction.

Motivation & Objective

• To generalize Fiedler's inequality μ₂(G) ≤ κ(G) from simple graphs to multigraphs.
• To define and analyze the role of edge multiplicity in bounding algebraic connectivity.
• To establish tightness of the bound μ₂(G) ≤ κ(G)·m(G) via explicit constructions of d-regular multigraphs.
• To investigate the relationship between algebraic connectivity, regularity, and vertex connectivity in multigraphs.
• To determine when equality μ₂(G) = d holds for d-regular multigraphs.

Proposed method

• Define m(G) as the maximum over all vertices v of the maximum edge multiplicity between v and any neighbor.
• Introduce κ(G) as the minimum vertex cut size, and μ₂(G) as the second smallest Laplacian eigenvalue.
• Use spectral graph theory and structural analysis to derive the inequality μ₂(G) ≤ κ(G)·m(G) for non-complete multigraphs with duplicated edges.
• Construct infinite families of d-regular multigraphs with μ₂(H) = d, κ(H) = t, and m(H) = d/t to demonstrate tightness of the bound.
• Analyze the implications of high algebraic connectivity (μ₂(G) > d/4) for 2-connectivity in d-regular multigraphs.
• Exclude complete graphs with duplicated edges as exceptional cases in the main inequality due to structural uniqueness.

Experimental results

Research questions

• RQ1 Does the inequality μ₂(G) ≤ κ(G) hold for multigraphs when adjusted for edge multiplicity?
• RQ2 Can the bound μ₂(G) ≤ κ(G)·m(G) be achieved with equality in multigraphs?
• RQ3 What structural conditions force μ₂(G) > d/4 to imply 2-connectivity in d-regular multigraphs?
• RQ4 For which d-regular multigraphs does equality μ₂(G) = d hold?
• RQ5 How do edge multiplicities affect the spectral gap and connectivity in multigraphs?

Key findings

• For all multigraphs G except those obtained by duplicating edges in a complete graph, μ₂(G) ≤ κ(G)·m(G), establishing a generalized Fiedler-type bound.
• The bound μ₂(G) ≤ κ(G)·m(G) is sharp, as demonstrated by constructing d-regular multigraphs with μ₂(H) = d, κ(H) = t, and m(H) = d/t for any t ≥ 2 and infinitely many d.
• For d-regular multigraphs (excluding edge-duplicated complete graphs), μ₂(G) ≤ d, with equality achieved in the constructed families.
• If μ₂(G) > d/4 in a d-regular multigraph (excluding the 2-vertex case), then G is 2-connected, linking high algebraic connectivity to strong connectivity.
• The maximum edge multiplicity per vertex, m(G), plays a critical role in modulating the upper bound of algebraic connectivity in multigraphs.
• The construction of d-regular multigraphs with μ₂(H) = d and m(H) = d/t shows that the multiplicative factor m(G) cannot be replaced by a smaller function of connectivity.