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

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 * = β 1/2 + β −1/2 ≈ 2.01980 and β 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, ∞).We also prove that the same conclusion about forbidden subgraph characterization holds for signed graphs, and we give an application to symmetric integer matrices whose diagonal entries are all zero. 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 R d where all pairwise inner products lie in {α, β} with −1 ≤ β < 0 ≤ α < 1. Very recently Jiang, Tidor, Yao, Zhang and Zhao determined the limit of N α,β (d)/d as d → ∞ when α + 2β < 0 or (1−α)/(α−β) ∈ {1, √ 2, √ 3}, 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 * .

Spherical Two-Distance Sets and Eigenvalues of Signed Graphs

“…Remark After this work is completed, building on our forbidden induced subgraph framework, Jiang and Polyanskii [8] proved Conjecture 1.11 for every λ < λ * , where λ * = β 1/2 + β −1/2 ≈ 2.01980 and β is the unique real root of…”

On signed graphs whose spectral radius does not exceed $\sqrt{2+\sqrt{5}}$

“…If i 2 = 5 or i 2 ≥ 6, then Ṫ7 ⊂ Ġ or Ṫ4 ⊂ Ġ, respectively, contradiction. So, i 2 = 4 and Ġ ⊂ [G 10 5 , v 10 , s]. Subcase 1.4.…”

“…, where i 2 = i 1 + 1 and v i 1 − ∼ v i 2 . By subcases 1.1 and 1.3, then Ġ ⊂ [G 10 5 , v 10 , s, v, Ġ], where the pair ( Ġ, v) ∈ {(P 4 , v 2 ), (Q 1,0 , v 1 ), (T a,1,a−1 , v a−1 ), (Q n 1 ,n 1 , v n 1 )} (a ≥ 3 and n 1 ≥ 1).…”