×

zbMATH — the first resource for mathematics

On the sum of two largest eigenvalues of a symmetric matrix. (English) Zbl 1148.05047
Summary: 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, 329–336 (2006; Zbl 1142.05343)], 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.

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] D. Gernert, private communication, see also <http://www.sgt.pep.ufrj.br/home_arquivos/prob_abertos.html>.
[2] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press Cambridge · Zbl 0576.15001
[3] B. Mohar, On the sum of k largest eigenvalues of a symmetric matrix, J. Combin. Theory Ser. B, in press. · Zbl 1217.05151
[4] Nikiforov, V., Linear combinations of graph eigenvalues, Electron. J. linear algebra, 15, 329-336, (2006) · Zbl 1142.05343
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.