A Geometric View of Combinatorial Fiedler Theory

2026-07-01

AI Summary

This study addresses the optimization problem associated with the largest Laplacian eigenvalue in the ℓ₁ sense within the combinatorial Fiedler theory of graphs, specifically the maximization of the parameter B(G). By unifying this problem within a geometric optimization framework, the authors reveal that the feasible set corresponds to an (n−2)-dimensional truncated octahedral shell. Their main contributions include deriving an exact analytical expression for B(G)—equal to the average of the two largest vertex degrees in the graph—proving that optimal solutions occur at polyhedral vertices while minimizers lie at face centers, and establishing that computing the number of vectors achieving B(G) is a #P-complete problem. This work bridges graph theory, combinatorial optimization, convex geometry, and computational complexity, uncovering deep connections between high-dimensional polyhedral structures and spectral graph properties.

Abstract

Recently, Andrade and Dahl introduced combinatorial Fiedler theory by studying a parameter $b(G)$ defined as the $\ell_1$-analog of the Rayleigh quotient minimization characterization of the algebraic connectivity of a graph $G=(V,E)$. In this work, we study the corresponding maximization problem, which plays the role of the $\ell_1$-analog of the largest Laplacian eigenvalue. We show that the new parameter $B(G)$ associated with this maximization problem admits a simple exact description: it is the average of the two largest vertex degrees of $G$. A unified combinatorial treatment of the minimization and maximization problems is presented first. Later, both optimization problems are reinterpreted in a geometrical setting. The feasible set is identified with a $(n-2)$-dimensional cuboctahedron shell where $n=|V|$. Additional structure is presented for this polyhedron, including the fact that maximizing solutions arise at its vertices and minimizing solutions arise at the centers of its facets. Finally, we analyze the number of optimal vectors for $b(G)$ and $B(G)$ for several graph families. Although the value of $B(G)$ is determined by the two largest degrees, we prove that counting the vectors that attain this value is actually $\#\mathrm{P}$-complete.

Problem

• combinatorial Fiedler theory
• Laplacian eigenvalue
• maximization problem
• graph degree
• optimization complexity

Innovation

• combinatorial Fiedler theory
• Laplacian eigenvalue
• ℓ₁ optimization
• cuboctahedron geometry
• #P-completeness