1. Conjectured bounds for the sum of squares of positive eigenvalues of a graph.
- Author
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Elphick, Clive, Farber, Miriam, Goldberg, Felix, and Wocjan, Pawel
- Subjects
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SUM of squares , *EIGENVALUES , *GRAPHIC methods , *MATHEMATICAL bounds , *LOGICAL prediction - Abstract
A well known upper bound for the spectral radius of a graph, due to Hong, is that μ 1 2 ≤ 2 m − n + 1 if δ ≥ 1 . It is conjectured that for connected graphs n − 1 ≤ s + ≤ 2 m − n + 1 , where s + denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete q -partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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