A structure theory for signed graphs with fixed smallest eigenvalue

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.