Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult

TL;DR

This work extends the Cameron–Goethals–Seidel–Shult classification by completely classifying connected graphs with smallest eigenvalue in (-λ^*, -2), where λ^* ≈ 2.0198. It introduces the notions of augmented path extensions and maverick graphs, and proves a two-phase classification: (i) all sufficiently large graphs in G(λ^*)ackslash G(2) are augmented-path extensions of rooted graphs (rooted graphs arise as line graphs of bipartite, connected single-rooted H_r), and (ii) the remaining graphs—mavericks—are enumerated explicitly up to 19 vertices, totaling 4752. The authors combine forbidden-subgraph techniques with a linear-algebraic reduction and computer-assisted enumeration to achieve a finite, constructive catalog of 794 rooted families and 4752 mavericks, providing a deep structural view of the transition at -λ^*. They also explore a twisted-maverick regime and discuss the limits of the method beyond λ^*, outlining open problems, including classification for (-λ', -λ^*) and extensions to signed graphs.

Abstract

In 1976, Cameron, Goethals, Seidel, and Shult classified all the graphs whose smallest eigenvalue is at least $-2$ by relating such graphs to root systems that appear in the classification of semisimple Lie algebras. In this paper, extending their beautiful theorem, we give a complete classification of all connected graphs whose smallest eigenvalue lies in $(-λ^*, -2)$, where $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$, and $ρ$ is the unique real root of $x^3 = x + 1$. Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in $(-λ, -2)$ for any constant $λ> 2$.

Figures (12)

• Figure 1: $E_4$, $E_5$, $E_6$ and $E_n$.
• Figure 2: The augmented path extension $(F_R, \ell,)$.
• Figure 3: A graph $\hat{H}$ with petals and a schematic drawing of its line graph $L(\hat{H})$.
• Figure 4: Three extensions of a rooted graph $F_R$.
• Figure 5: Four $8$-vertex graphs $F$ with two vertices $v_6$ and $v_7$ such that $F - \left\{{v_6, v_7}\right\}$ is isomorphic to $E_6$, and both $F - v_6$ and $F - v_7$ are isomorphic to $E_7$.

Theorems & Definitions (79)

• Proposition 1.1: Hoffman H72
• Definition 1.2: Augmented path extension
• Definition 1.3: Maverick graph
• Theorem 1.4: Classification theorem at a glance
• Theorem 1.5
• Lemma 1.6
• Corollary 1.7
• Definition 2.1: Graph with petals and generalized line graph
• Theorem 2.2: Theorem 2.1 of Hoffman H77b
• Theorem 2.3: Cvetković, Doob, and Simić CDS80CDS81, and Rao, Singhi, and Vijayan RSV81