Abstract
To every symmetric matrixA with entries ±1, we associate a graph G(A), and ask (for two different definitions of distance) for the distance ofG(A) to the nearest complete bipartite graph (cbg). Letλ 1(A),λ 1 (A) be respectively the algebraically largest and least eigenvalues ofA. The Frobenius distance (see Section 4) to the nearest cbg is bounded above and below by functions ofn −λ 1 (A), wheren=ord A. The ordinary distance (see Section 1) to the nearest cbg is shown to be bounded above and below by functions ofλ 1 (A). A curious corollary is: there exists a functionf (independent ofn, and given by (1.1)), such that |λ i (A) | ≦f(λ 1(A), whereλ i (A) is any eigenvalue ofA other thanλ i (A).
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This work was supported (in part) by the U.S. Army under contract #DAHC04-C-0023.
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Hoffman, A.J. On eigenvalues of symmetric (+1, −1) matrices. Israel J. Math. 17, 69–75 (1974). https://doi.org/10.1007/BF02756827
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DOI: https://doi.org/10.1007/BF02756827