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Matroid -- from Wolfram MathWorld   Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book 12,902 entries Last updated: Mon Nov 10 2008   Foundations of Mathematics > Set Theory > Sets >Matroid Roughly speaking, a matroid is a finite set together with a generalization of a concept from linear algebra that satisfies a natural set of properties for that concept. For example, the finite set could be the rows of a matrix, and the generalizing concept could be linear dependence and independence of any subset of rows of the matrix.Formally, a matroid consists of a finite set  of elements together with a family  of nonempty subsets of  , called circuits, which satisfy the axioms 1. No proper subset of a circuitis a circuit, 2. If  and  , then  contains a circuit. (Harary 1994, p. 40).An equivalent definition considers a matroid as a finite set  of elements together with a family of subsets of  , called independent sets, such that 1. The empty set is independent, 2. Every subset of an independent setis independent, 3. For every subset  of  , all maximal independent sets contained in  have the same number of elements. (Harary 1994, pp. 40-41).The number of simple matroids (or combinatorial geometries) with  , 1, ... points are 1, 1, 2, 4, 9, 26, 101, 950, ... (Sloane's A002773), and the number of matroids on  , 1, ... points are 1, 2, 4, 8, 17, 38, 98, 306, 1724, ... (Sloane's A055545; Oxley 1993, p. 473). (The value for  given by Oxley 1993, p. 42, is incorrect.)SEE ALSO: Combinatorial Geometry, Graphoid, Oriented MatroidREFERENCES:Björner, A.; Las Vergnas, M.; Sturmfels, B.; White, N.; and Ziegler, G. Oriented Matroids, 2nd ed. Cambridge, England: Cambridge University Press, 1999.Blackburn, J. E.; Crapo, H. H.; and Higgs, D. A. "A Catalogueof Combinatorial Geometries." Math. Comput. 27, 155-166, 1973.Crapo, H. H. and Rota, G.-C. "On the Foundations of Combinatorial Theory. II. Combinatorial Geometries." Cambridge, MA: MIT Press, 109-133, 1970.Harary, F. "Matroids." Graph Theory. Reading, MA: Addison-Wesley, pp. 40-41, 1994.Minty, G. "On the Axiomatic Foundations of the Theories of Directed Linear Graphs, Electric Networks, and Network-Programming." J. Math. Mech. 15, 485-520, 1966.Oxley, J. G. Matroid Theory. Oxford, England: Oxford University Press, 1993.Papadimitriou, C. H. and Steiglitz, K. Combinatorial Optimization: Algorithms and Complexity. Englewood Cliffs, NJ: Prentice-Hall, 1982.Richter-Gebert, J. and Ziegler, G. M. In Handbook of Discrete and Computational Geometry (Ed. J. E. Goodman and J. O'Rourke). Boca Raton, FL: CRC Press, pp. 111-112, 1997.Sloane, N. J. A. Sequences A002773/M1197 and A055545 in "The On-Line Encyclopedia of Integer Sequences."Sloane, N. J. A. and Plouffe, S. Figure M1197 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.Tutte, W. T. "Lectures on Matroids." J. Res. Nat. Bur. Stand. Sect.B 69, 1-47, 1965.Whitely, W. "Matroids and Rigid Structures." In Matroid Applications, Encyclopedia of Mathematics and Its Applications (Ed. N. White), Vol. 40. New York: Cambridge University Press, pp. 1-53, 1992.Whitney, H. "On the Abstract Properties of Linear Dependence." Amer.J. Math. 57, 509-533, 1935.CITE THIS AS:Weisstein, Eric W. "Matroid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Matroid.html Contact the MathWorld Team© 1999-2008 Wolfram Research, Inc. | Terms of Use  Other Wolfram Web Resources » Wolfram Research Mathematica Home Page Mathematica Documentation Center Wolfram Demonstrations Project Online Integrator Functions Site Wolfram Tones Wolfram Science Wolfram Blog more… |
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