Tan–Koolen–Xia conjecture on co-edge-regular graphs with four eigenvalues

All graphs considered are finite, undirected and simple. A graph is co-edge-regular with parameters $(n,k,c)$ if it is $k$-regular on $n$ vertices and every two distinct non-adjacent vertices have exactly $c$ common neighbours. Write $\theta_{\min}(G)$ for the smallest eigenvalue of the adjacency matrix of $G$; an $s$-clique extension is the graph obtained by replacing each vertex by a clique of size $s$ according to the adjacency relations of the original graph. A $p \times q$-grid is the Cartesian product of two complete graphs, with $p>q\geqslant 2$. Let $G$ be a connected co-edge-regular graph with parameters $(n,k,c)$ having four distinct eigenvalues, and let $m\geqslant 2$ be an integer. Tan–Koolen–Xia conjecture. There exists a constant $n_m$ such that, if

$$\theta_{\min}(G)\geqslant -m,\qquad n\geqslant n_m,\qquad k<n-2-\frac{(m-1)^2}{4},$$

then either $G$ is the $s$-clique extension of a strongly regular graph for some $2\leqslant s\leqslant m-1$, or $G$ is a $p\times q$-grid with $p>q\geqslant 2$. This conjecture predicts a classification of sufficiently large connected co-edge-regular graphs with four distinct eigenvalues under a lower bound on the smallest eigenvalue and an upper bound on the valency.