Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Maximum spread of graphs and bipartite graphs   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.

20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex