2022/06/08 by Liu, Lele
#05C50 #15A18 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2206.03723
We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let λ1(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 λ1(G) + λ1(G) is the join of a clique and an independent set, with \lfloor n/3\rfloor and \lceil 2n/3\rceil (also \lceil n/3\rceil and \lfloor 2n/3\rfloor if n ≡ 2 \pmod3) vertices, respectively. We resolve this conjecture for sufficiently large n using analytic methods. Our second result concerns the Q-spread sQ(G) 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 Cvetković, Rowlinson and Simić in 2007 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.