Maximum spread of graphs and bipartite graphs

Abstract

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 2n/3 and n/3 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 L 2 [0, 1]. The second conjecture claims that for any fixed e ≤ 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.

On the maximum spread of planar and outerplanar graphs

“…One of the central focuses of this area is to find the maximum or minimum spread over a fixed family of graphs and characterize the extremal graphs. Problems of such extremal flavor have been investigated for trees [1], graphs with few cycles [12,18,27], the family of all n-vertex graphs [2,4,20,22,23,25], the family of bipartite graphs [4], graphs with a given matching number [16], girth [26], or size [15], and very recently for the family of outerplanar graphs [13]. We note that the spreads of other matrices associated with a graph have also been extensively studied (see e.g.…”

On the spread of outerplanar graphs

“…A problem in this area with an extremal flavor is to maximize or minimize the spread over a fixed family of graphs. This problem has been considered for trees [1], graphs with few cycles [13,27,38], the family of all n-vertex graphs [3,6,29,31,32,35], bipartite graphs [6], graphs with a given matching number [19] or girth [36] or size [22].…”