Forbidden induced subgraphs for graphs and signed graphs with eigenvalues bounded from below

TL;DR

The paper investigates when graphs with smallest eigenvalue at least $-\lambda$ admit a finite forbidden induced subgraph characterization. It proves a sharp threshold: such a characterization exists if and only if $\lambda<\lambda^*\approx 2.01980$, with $\lambda^* = \rho^{1/2}+\rho^{-1/2}$ and $\rho$ the unique real root of $x^3=x+1$; it also extends the threshold to signed graphs. The authors develop a robust framework combining Hoffman-type interlacing, extension families, and Dickson’s lemma to certify finiteness for $\lambda<\lambda^*$ and identify structural forcing to generalized line graphs for $\lambda\in[2,\lambda^*)$, thereby ruling out infinite families of forbidden subgraphs beyond the threshold. As a byproduct, they characterize limit points of smallest eigenvalues and connect these results to questions on spherical two-distance sets via signed graphs. The work resolves a longstanding question of Bussemaker–Neumaier and provides a precise bridge between spectral graph theory and high-dimensional geometry.

Abstract

The smallest eigenvalue of a graph is the smallest eigenvalue of its adjacency matrix. We show that the family of graphs with smallest eigenvalue at least $-λ$ can be defined by a finite set of forbidden induced subgraphs if and only if $λ< λ^*$, where $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$, and $ρ$ is the unique real root of $x^3 = x + 1$. This resolves a question raised by Bussemaker and Neumaier. As a byproduct, we find all the limit points of smallest eigenvalues of graphs, supplementing Hoffman's work on those limit points in $[-2, \infty)$. We also prove that the same conclusion about forbidden subgraph characterization holds for signed graphs. Our impetus for the study of signed graphs is to determine the maximum cardinality of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Denote by $N_{α, β}(n)$ the maximum number of unit vectors in $\mathbb{R}^d$ where all pairwise inner products lie in $\{α, β\}$ with $-1 \le β< 0 \le α< 1$. Very recently Jiang, Tidor, Yao, Zhang and Zhao determined the limit of $N_{α, β}(d)/d$ as $d\to\infty$ when $α+ 2β< 0$ or $(1-α)/(α-β) \in \{1,\sqrt2,\sqrt3\}$, and they proposed a conjecture on the limit in terms of eigenvalue multiplicities of signed graphs. We establish their conjecture whenever $(1-α)/(α- β) < λ^*$.

Figures (4)

• Figure 1: Maximal connected graphs with spectral radius at most $2$. The number of vertices is one more than the given index. In particular, $\widetilde{D}_4$ is actually a star with four leaves.
• Figure 2: $E_{2,n}$
• Figure 3: The claw graph $C$ and the diamond graph $D$.
• Figure :

Theorems & Definitions (27)

• Remark
• Definition 1.1
• Theorem 1.2: Theorem 1 of Jiang and Polyanskii JP20
• Theorem 1.3
• Remark
• Corollary 1.4
• Theorem 1.5
• Corollary 1.6
• Definition 1.7
• Conjecture 1.8: Conjecture 1.11 of Jiang et al. JTYZZ23