405,914 research outputs found
Miscellaneous Graph Preliminaries
This article contains many auxiliary theorems which were missing in the Mizar Mathematical Library [2] to the best of the author’s knowledge. Most of them regard graph theory as formalized in the GLIB series (cf. [8]) and most of them are preliminaries needed in [7] or other forthcoming [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.John Adrian Bondy and U. S. R. Murty. Graph Theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. ISBN 978-1-84628-969-9.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. About graph mappings. Formalized Mathematics, 27(3):261–301, 2019. doi:10.2478/forma-2019-0024.Sebastian Koch. About graph complements. Formalized Mathematics, 28(1):41–63, 2020. doi:10.2478/forma-2020-0004.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.233
About Vertex Mappings
In [6] partial graph mappings were formalized in the Mizar system [3]. Such mappings map some vertices and edges of a graph to another while preserving adjacency. While this general approach is appropriate for the general form of (multidi)graphs as introduced in [7], a more specialized version for graphs without parallel edges seems convenient. As such, partial vertex mappings preserving adjacency between the mapped verticed are formalized [email protected] Gutenberg University, Mainz, GermanyIan Anderson. A first course in discrete mathematics. Springer Undergraduate Mathematics Series. Springer, London, 2001. ISBN 1-85233-236-0.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. About graph mappings. Formalized Mathematics, 27(3):261–301, 2019. doi:10.2478/forma-2019-0024.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.27330331
About Graph Mappings
In this articles adjacency-preserving mappings from a graph to another are formalized in the Mizar system [7], [2]. The generality of the approach seems to be largely unpreceeded in the literature to the best of the author’s knowledge. However, the most important property defined in the article is that of two graphs being isomorphic, which has been extensively studied. Another graph decorator is introduced as [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.John Adrian Bondy and U. S. R. Murty. Graph Theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. ISBN 978-1-84628-969-9.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1): 153–164, 1990.Christopher David Godsil and Gordon Royle. Algebraic graph theory. Graduate Texts in Mathematics; 207. Springer, New York, 2001. ISBN 0-387-95220-9; 0-387-95241-1.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Huckenbeck. Extremal paths in graphs: foundations, search strategies, and related topics, volume 10 of Mathematical Topics. Akademie Verlag, Berlin, 1. edition, 1997. ISBN 3-05-501658-0; 978-3-05-501658-5.Tommy R. Jensen and Bjarne Toft. Graph coloring problems. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York, 1995. ISBN 0-471-02865-7.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. About supergraphs. Part I. Formalized Mathematics, 26(2):101–124, 2018. doi:10.2478/forma-2018-0009.Sebastian Koch. About supergraphs. Part II. Formalized Mathematics, 26(2):125–140, 2018. doi:10.2478/forma-2018-0010.Mike Krebs and Anthony Shaheen. Expander families and Cayley graphs: a beginners guide. Oxford University Press, Oxford, 2011. ISBN 0-19-976711-4; 978-0-19-976711-3.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.27326130
Unification of Graphs and Relations in Mizar
A (di)graph without parallel edges can simply be represented by a binary relation of the vertices and on the other hand, any binary relation can be expressed as such a graph. In this article, this correspondence is formalized in the Mizar system [2], based on the formalization of graphs in [6] and relations in [11], [12]. Notably, a new definition of createGraph will be given, taking only a non empty set V and a binary relation E ⊆ V × V to create a (di)graph without parallel edges, which will provide to be very useful in future [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. Underlying simple graphs. Formalized Mathematics, 27(3):237–259, 2019. doi:10.2478/forma-2019-0023.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Karol Pąk. The friendship theorem. Formalized Mathematics, 20(3):235–237, 2012. doi:10.2478/v10037-012-0028-7.Gunther Schmidt and Thomas Ströhlein. Relations and graphs: discrete mathematics for computer scientists. Springer Science & Business Media, 2012.Yozo Toda. The formalization of simple graphs. Formalized Mathematics, 5(1):137–144, 1996.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.28217318
Extended Natural Numbers and Counters
This article introduces extended natural numbers, i.e. the set ℕ ∪ {+∞}, in Mizar [4], [3] and formalizes a way to list a cardinal numbers of cardinals. Both concepts have applications in graph [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.John Adrian Bondy and U. S. R. Murty. Graph Theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. ISBN 978-1-84628-969-9.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Library Committee of the Association of Mizar Users. Number-valued functions. Mizar Mathematical Library, 2007.Library Committee of the Association of Mizar Users. Introduction to arithmetic of extended real numbers. Mizar Mathematical Library, 2006.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341–347, 2003.Andrzej Trybulec. Subsets of complex numbers. Mizar Mathematical Library, 2003.Andrzej Trybulec. Basic operations on extended real numbers. Mizar Mathematical Library, 2008.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825–829, 2001.28323924
A graph-based mathematical morphology reader
This survey paper aims at providing a "literary" anthology of mathematical
morphology on graphs. It describes in the English language many ideas stemming
from a large number of different papers, hence providing a unified view of an
active and diverse field of research
Trivalent graphs, volume conjectures and character varieties
The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum
volume conjecture) are extended to U_q(\fraksl_2) colored quantum invariants
of the theta and tetrahedron graph. The \SL(2,\bC) character variety of the
fundamental group of the complement of a trivalent graph with edges in
is a Lagrangian subvariety of the Hitchin moduli space over the Riemann
surface of genus . For the theta and tetrahedron graph, we conjecture
that the configuration of the character variety is locally determined by large
color asymptotics of the quantum invariants of the trivalent graph in terms of
complex Fenchel-Nielsen coordinates. Moreover, the -holonomic difference
equation of the quantum invariants provides the quantization of the character
variety.Comment: 11 pages, 2 figure
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
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