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Brouwer has confirmed by computation that the conjecture is valid for all graphs with at most 10 vertices. [1] It is also known that the conjecture is valid for any number of vertices if t = 1, 2, n − 1, and n. For certain types of graphs, Brouwer's conjecture is known to be valid for all t and for any number of vertices
Although Brouwer writes that this graph's "construction is folklore", and cites as an early reference a 1964 paper on Latin squares by Dale M. Mesner, [1] it is named after Andries Brouwer and Willem H. Haemers, who in 1992 published a proof that it is the only strongly regular graph with the same parameters. [3]
Andries Brouwer and Hendrik van Maldeghem (see #References) use an alternate but fully equivalent definition of a strongly regular graph based on spectral graph theory: a strongly regular graph is a finite regular graph that has exactly three eigenvalues, only one of which is equal to the degree k, of multiplicity 1.
The 1980 monograph Spectra of Graphs [16] by Cvetković, Doob, and Sachs summarised nearly all research to date in the area. In 1988 it was updated by the survey Recent Results in the Theory of Graph Spectra. [17] The 3rd edition of Spectra of Graphs (1995) contains a summary of the further recent contributions to the subject. [15]
The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that n is even. If there were a fixed-point-free continuous self-mapping f of the closed unit ball B of the n -dimensional Euclidean space V , set
Andries Evert Brouwer (born 1951) is a Dutch mathematician and computer programmer, Professor Emeritus at Eindhoven University of Technology (TU/e). He is known as the creator of the greatly expanded 1984 to 1985 versions of the roguelike computer game Hack that formed the basis for NetHack . [ 1 ]
The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to topological vector spaces, which may be of infinite dimension.It asserts that if is a nonempty convex closed subset of a Hausdorff topological vector space and is a continuous mapping of into itself such that () is contained in a compact subset of , then has a fixed point.
Kolmogorov supports most of Brouwer's results but disputes a few; he discusses the ramifications of intuitionism with respect to "transfinite judgements", e.g. transfinite induction. 1927. L. E. J. Brouwer: "On the domains of definition of functions". Brouwer's intuitionistic treatment of the continuum, with an extended commentary. 1927.