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 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 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 pendant edge to Kn-1. We confirm this conjecture for sufficiently large n.