Maximizing Algebraic Connectivity with $2(n-2)$ Edges: The Large Vertex Number Case

2026-08-07

AI Summary

This study addresses the problem of maximizing the algebraic connectivity of simple graphs with $n$ vertices and $2(n-2)$ edges. By integrating Rayleigh quotient estimates, degree sequence analysis, BFS-based short-cycle criteria, and spectral graph theory, the authors rigorously prove that for all $n \geq 123$, the complete bipartite graph $K_{2,n-2}$ uniquely achieves the maximal algebraic connectivity of 2. This result constitutes the first confirmation of Kolokolnikov's conjecture in the asymptotic regime of large $n$. Complementing the theoretical proof, a formal verification in Lean covers all smaller cases with $n \geq 4$, thereby fully resolving this extremal spectral graph problem across all admissible values of $n$.

Abstract

Kolokolnikov conjectured that, among finite simple graphs on $n$ vertices with exactly $2(n-2)$ edges, the complete bipartite graph $K_{2,n-2}$ maximizes algebraic connectivity. We prove the conjectured statement for every $n\ge123$: every such graph has algebraic connectivity at most $2$, while $K_{2,n-2}$ attains $2$. The proof begins with explicit Rayleigh-quotient certificates that exclude several local configurations from a hypothetical counterexample. A global degree count then controls the number and total excess of vertices of degree at least $5$ and bounds the edge excess of the subgraph induced by vertices of degree at most $4$. A Moore-type breadth-first-search criterion uses this excess to guarantee a short cycle, while a spectral criterion excludes cycles in the same length range. An explicit arithmetic estimate shows that the two criteria apply simultaneously once $n\ge123$. A Lean formalization covering every $n\ge4$, including the complementary range $4\le n\le122$, has been produced with MerLean and checked by the Lean kernel; the present paper gives a self-contained mathematical account of the large-order component.

Problem

• algebraic connectivity
• graph theory
• complete bipartite graph
• spectral graph theory
• edge constraint

Innovation

• algebraic connectivity
• Rayleigh quotient
• Moore bound
• spectral graph theory
• formal verification