Sesqui-regular graphs with fixed smallest eigenvalue

Let $\lambda \ge 2$ be an integer. For strongly regular graphs with parameters $(v,k,a,c)$ and smallest eigenvalue $-\lambda$, Neumaier gave two bounds on $c$ by using algebraic property of strongly regular graphs. In this paper, we will study a new class of regular graphs called sesqui-regular graphs, which contains strongly regular graphs as a subclass, and prove that for a sesqui-regular graph with parameters $(v,k,c)$ and smallest eigenvalue at least $-\lambda$, if $k$ is very large, then either $c\le {\lambda }^2(\lambda -1)$ or $v-k-1\le \frac{(\lambda -1{)}^2}{4}+1$ holds.