  Let $\lambda\geq2$ 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 $c \leq \lambda^2(\lambda -1)$ or
$v-k-1 \leq \frac{(\lambda-1)^2}{4} + 1$ holds.
