Spectral Extrema for Graphs: The Zarankiewicz Problem

Abstract

Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővári-Turán-Sós upper bound on the average degree of $G$ (the Zarankiewicz problem).