2026/07/17 by Sahil Agarwal, Carter Antley, Joseph Aulenbacher +6
#math.CO
For a graph G, let λ1(G)≥ λ2(G)≥ ⋯ ≥ λn(G) denote the adjacency eigenvalues of G. We investigate the asymptotic maximum of λi(G)+λj( G) for fixed i and j. We prove general bounds on λi(G) + λj(G) for all pairs (i, j) and also give general bounds on the related problem of minimizing λn-i+1(G) + λn-j+1(G) for fixed i and j. We prove that for all looped graphs G on n vertices, λ1(G) + λ2(G) ≤ \frac87 n. Our method also gives a new short proof of the Nordhaus-Gaddum result for the spectral radius proved by Terpai that λ1(G) + λ1(G) ≤ \frac43n - 1. We also show the close relation of these Nordhaus-Gaddum type problems to recent work on the maximum spectral gaps of graphs by Brooks, Linz and Lu.