Graph Limits and Spectral Extremal Problems for Graphs

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 590-608, March 2024.
Abstract. We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let [math] be the largest eigenvalue of the adjacency matrix of a graph [math] and [math] be the complement of [math]. A nice conjecture states that the graph on [math] vertices maximizing [math] is the join of a clique and an independent set with [math] and [math] (also [math] and [math] if [math]) vertices, respectively. We resolve this conjecture for sufficiently large [math] using analytic methods. Our second result concerns the [math]-spread of a graph [math], which is defined as the difference between the largest eigenvalue and least eigenvalue of the signless Laplacian of [math]. It was conjectured by Cvetković, Rowlinson, and Simić [Publ. Inst. Math., 81 (2007), pp. 11–27] that the unique [math]-vertex connected graph of maximum [math]-spread is the graph formed by adding a pendant edge to [math]. We confirm this conjecture for sufficiently large [math].