The Largest Eigenvalue and Bi-Average Degree of a Graph

We show that for a graph $G$ with the vertex set $V$ and the largest eigenvalue $\lambda_{\max}(G)$, letting $$M(G) := \max_{X,Y \subset V} \frac{e(X,Y)}{\sqrt{|X||Y|}}$$ (where $e(X,Y)$ denotes the number of edges between $X$ and $Y$), we have $$M(G) \le \lambda_{\max}(G) \le \big(\frac14 \log|V| + 1 \big) \M(G).$$ Here the lower bound is attained if $G$ is regular or bi-regular, whereas the logarithmic factor in the upper bound, conjecturally, can be improved --- although we present an example showing that it cannot be replaced with a factor growing slower than $(\log |V|/\log\log|V|)^{1/8}$. Further refinements are established, particularly in the case where $G$ is bipartite.