A structure theory for signed graphs with fixed smallest eigenvalue

Abstract

In this paper, we will give a structure theory for signed graphs with fixed smallest eigenvalue and investigate signed graphs with smallest eigenvalue greater than $-1-\sqrt{2}$. Given a real number $λ\leq -1$, we show that the following hold for each signed graph $(G,σ)$ with smallest eigenvalue at least $λ$ and large minimum valency: $\mathrm{(i)}$ there exist dense induced subgraphs $N_1, \dots, N_r$ in $(G,σ)$ such that each vertex lies in at most $\lfloor -λ\rfloor$ $N_i$'s and almost all edges of $(G,σ)$ lie in at least one of the $N_i$'s; $\mathrm{(ii)}$ if $λ>-1-\sqrt{2}$, then $(G,σ)$ has smallest eigenvalue at least $-2$ and $(G,σ)$ is $1$-integrable.

Figures (1)

• Figure 1: The signed graphs of $\widetilde{K_4}^{(0)}$ and $\widetilde{K_4}^{(-)}$

Theorems & Definitions (54)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5: Gavrilyuk2021signed
• Theorem 1.6
• Remark 1.7
• Theorem 2.1: ramsey1930
• Lemma 2.2
• proof