Article ID Journal Published Year Pages File Type
4603283 Linear Algebra and its Applications 2008 7 Pages PDF
Abstract

Gernert conjectured that the sum of two largest eigenvalues of the adjacency matrix of any simple graph is at most the number of vertices of the graph. This can be proved, in particular, for all regular graphs. Gernert’s conjecture was recently disproved by one of the authors [V. Nikiforov, Linear combinations of graph eigenvalues, Electron. J. Linear Algebra 15 (2006) 329–336], who also provided a nontrivial upper bound for the sum of two largest eigenvalues. In this paper we improve the lower and upper bounds to near-optimal ones, and extend results from graphs to general non-negative matrices.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory