  Given any graph $G$, the (adjacency) spread of $G$ is the maximum absolute
difference between any two eigenvalues of the adjacency matrix of $G$. 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 $n$, the $n$-vertex graph $G$ that maximizes spread is the
join of a clique and an independent set, with $\lfloor 2n/3 \rfloor$ and
$\lceil n/3 \rceil$ vertices, respectively. Using techniques from the theory of
graph limits and numerical analysis, we prove this claim for all $n$
sufficiently large. As an intermediate step, we prove an analogous result for a
family of operators in the Hilbert space over $\mathscr{L}^2[0,1]$. The second
conjecture claims that for any fixed $e\leq n^2/4$, if $G$ maximizes spread
over all $n$-vertex graphs with $e$ edges, then $G$ is bipartite. We prove an
asymptotic version of this conjecture. Furthermore, we exhibit an infinite
family of counterexamples, which shows that our asymptotic solution is tight up
to lower order error terms.
