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Given any graph G G , the spread of G G is the maximum difference between any two eigenvalues of the adjacency matrix of G G . In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers n n , the n n -vertex graph G G that maximizes spread is the join of a clique and an independent set, with ⌊ 2 n / 3 ⌋ \lfloor 2n/3 \rfloor and ⌈ n / 3 ⌉ \lceil n/3 \rceil vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all n n sufficiently large. As an intermediate step, we prove an analogous result for a family of operators in the Hilbert space over L 2 [ 0 , 1 ] \mathscr {L}^2[0,1] . The second conjecture claims that for any fixed m ≤ n 2 / 4 m \leq n^2/4 , if G G maximizes spread over all n n -vertex graphs with m m edges, then G G is bipartite. We prove an asymptotic version of this conjecture. Furthermore, we construct an infinite family of counterexamples, which shows that our asymptotic solution is tight up to lower-order error terms.

We define the discrete norm of a complex $m\times n$ matrix $A$ by $$ \|A\|_\Delta := \max_{0\ne\xi\in\{0,1\}^n} \frac{\|A\xi\|}{\|\xi\|}, $$ and show that $$ \frac c{\sqrt{\log h(A)+1}}\,\|A\| \le \|A\|_\Delta \le \|A\|, $$ where $c>0$ is an explicitly indicated absolute constant, $h(A)=\sqrt{\|A\|_1\|A\|_\infty}/\|A\|$, and $\|A\|_1,\|A\|_\infty$, and $\|A\|=\|A\|_2$ are the induced operator norms of $A$. Similarly, for the \emph{discrete Rayleigh norm} $$ \|A\|_P := \max_{\substack{0\ne\xi\in\{0,1\}^m \\ 0\ne\eta\in\{0,1\}^n}} \frac{|\xi^tA\eta|}{\|\xi\|\|\eta\|} $$ we prove the estimate $$ \frac c{\log h(A)+1}\,\|A\| \le \|A\|_P \le \|A\|. $$ These estimates are shown to be essentially best possible. As a consequence, we obtain another proof of the (slightly sharpened and generalized version of the) converse to the expander mixing lemma by Bollobas-Nikiforov and Bilu-Linial.