GRAPH LIMITS AND SPECTRAL EXTREMAL PROBLEMS FOR GRAPHS

We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let \lambda1(G) be the largest eigenvalue of the adjacency matrix of a graph G and G be the complement of G. A nice conjecture states that the graph on n vertices maximizing \lambda1(G)+ \lambda1(G) is the join of a clique and an independent set with Ln/3\rfloor and [2n/3\rceil (also [n/3\rceil and L2n/3\rfloor if n \equiv 2 (mod 3)) vertices, respectively. We resolve this conjecture for sufficiently large n using analytic methods. Our second result concerns the Q -spread of a graph G, which is defined as the difference between the largest eigenvalue and least eigenvalue of the signless Laplacian of G. It was conjectured by Cvetkovic'\, Rowlinson, and Simic '\ [Publ. Inst. Math., 81 (2007), pp. 1127] that the unique n -vertex connected graph of maximum Q -spread is the graph formed by adding a pendant edge to Kn-1. We confirm this conjecture for sufficiently large n.