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