A structure theory for signed graphs with fixed smallest eigenvalue

TL;DR

This paper develops a structure theory for signed graphs with a fixed smallest eigenvalue, revealing dense subgraph configurations and properties related to eigenvalue bounds and integrability.

Contribution

It introduces a new structure theory for signed graphs with fixed smallest eigenvalue and characterizes graphs with eigenvalues greater than 1-22, including their integrability.

Findings

Existence of dense induced subgraphs covering most edges.

Graphs with eigenvalue > 1-22 are 2-integrable.

Graphs with eigenvalue > 1-22 have smallest eigenvalue at least 2-2.

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 $\lambda\leq -1$, we show that the following hold for each signed graph $(G,\sigma)$ with smallest eigenvalue at least $\lambda$ and large minimum valency: $\mathrm{(i)}$ there exist dense induced subgraphs $N_1, \dots, N_r$ in $(G,\sigma)$ such that each vertex lies in at most $\lfloor -\lambda\rfloor$ $N_i$'s and almost all edges of $(G,\sigma)$ lie in at least one of the $N_i$'s; $\mathrm{(ii)}$ if $\lambda>-1-\sqrt{2}$, then $(G,\sigma)$ has smallest eigenvalue at least $-2$ and $(G,\sigma)$ is $1$-integrable.