Abstract
Let $H$ be a graph and $t\geq s\geq 2$ be integers. We prove that if $G$ is
an $n$-vertex graph with no copy of $H$ and no induced copy of $K_{s,t}$, then
$\lambda(G) = O\left(n^{1-1/s}\right)$ where $\lambda(G)$ is the spectral
radius of the adjacency matrix of $G$. Our results are motivated by results of
Babai, Guiduli, and Nikiforov bounding the maximum spectral radius of a graph
with no copy (not necessarily induced) of $K_{s,t}$.