The integrability of sesqui-regular graphs with smallest eigenvalue at least $-3$

In this paper, we prove that every connected sesqui-regular graph with parameters $(n,k,c)$, where $c\ge 3$, and with smallest eigenvalue in $[-3,-2)$ is $1$-integrable if its valency is sufficiently large. The bound on $c$ is optimal: for every integer $t\ge 2$, the Cartesian product of the Shrikhande graph and $K_t$ is a connected sesqui-regular graph with parameters $(16t,t+5,2)$ and smallest eigenvalue $-3$, but it is not $1$-integrable.