MD - Matemàtica Discreta
http://hdl.handle.net/2117/3546
2017-01-18T12:01:48ZBounding the gap between a free group (outer) automorphism and its inverse
http://hdl.handle.net/2117/98389
Bounding the gap between a free group (outer) automorphism and its inverse
Ladra, Manuel; Silva, Pedro V.; Ventura Capell, Enric
For any finitely generated group GG , two complexity functions aGaG and ßGßG are defined to measure the maximal possible gap between the norm of an automorphism (respectively, outer automorphism) of GG and the norm of its inverse. Restricting attention to free groups FrFr , the exact asymptotic behaviour of a2a2 and ß2ß2 is computed. For rank r¿3r¿3 , polynomial lower bounds are provided for arar and ßrßr , and the existence of a polynomial upper bound is proved for ßrßr .
The final publication is available at Springer via http://dx.doi.org/10.1007/s13348-015-0133-3.
2016-12-15T18:01:37ZLadra, ManuelSilva, Pedro V.Ventura Capell, EnricFor any finitely generated group GG , two complexity functions aGaG and ßGßG are defined to measure the maximal possible gap between the norm of an automorphism (respectively, outer automorphism) of GG and the norm of its inverse. Restricting attention to free groups FrFr , the exact asymptotic behaviour of a2a2 and ß2ß2 is computed. For rank r¿3r¿3 , polynomial lower bounds are provided for arar and ßrßr , and the existence of a polynomial upper bound is proved for ßrßr .Algorithmic recognition of infinite cyclic extensions
http://hdl.handle.net/2117/96690
Algorithmic recognition of infinite cyclic extensions
Cavallo, Bren; Delgado Rodríguez, Jordi; Kahrobaei, Delaram; Ventura Capell, Enric
We prove that one cannot algorithmically decide whether a finitely presented Z-extension admits a finitely generated base group, and we use this fact to prove the undecidability of the BNS invariant. Furthermore, we show the equivalence between the isomorphism problem within the subclass of unique Z-extensions, and the semi-conjugacy problem for deranged outer automorphisms.
2016-11-15T16:32:01ZCavallo, BrenDelgado Rodríguez, JordiKahrobaei, DelaramVentura Capell, EnricWe prove that one cannot algorithmically decide whether a finitely presented Z-extension admits a finitely generated base group, and we use this fact to prove the undecidability of the BNS invariant. Furthermore, we show the equivalence between the isomorphism problem within the subclass of unique Z-extensions, and the semi-conjugacy problem for deranged outer automorphisms.Put three and three together: Triangle-driven community detection
http://hdl.handle.net/2117/89696
Put three and three together: Triangle-driven community detection
Prat Pérez, Arnau; Domínguez Sal, David; Brunat Blay, Josep Maria; Larriba Pey, Josep
Community detection has arisen as one of the most relevant topics in the field of graph data mining due to its applications in many fields such as biology, social networks, or network traffic analysis. Although the existing metrics used to quantify the quality of a community work well in general, under some circumstances, they fail at correctly capturing such notion. The main reason is that these metrics consider the internal community edges as a set, but ignore how these actually connect the vertices of the community. We propose the Weighted Community Clustering (WCC), which is a new community metric that takes the triangle instead of the edge as the minimal structural motif indicating the presence of a strong relation in a graph. We theoretically analyse WCC in depth and formally prove, by means of a set of properties, that the maximization of WCC guarantees communities with cohesion and structure. In addition, we propose Scalable Community Detection (SCD), a community detection algorithm based on WCC, which is designed to be fast and scalable on SMP machines, showing experimentally that WCC correctly captures the concept of community in social networks using real datasets. Finally, using ground-truth data, we show that SCD provides better quality than the best disjoint community detection algorithms of the state of the art while performing faster.
2016-09-08T08:14:33ZPrat Pérez, ArnauDomínguez Sal, DavidBrunat Blay, Josep MariaLarriba Pey, JosepCommunity detection has arisen as one of the most relevant topics in the field of graph data mining due to its applications in many fields such as biology, social networks, or network traffic analysis. Although the existing metrics used to quantify the quality of a community work well in general, under some circumstances, they fail at correctly capturing such notion. The main reason is that these metrics consider the internal community edges as a set, but ignore how these actually connect the vertices of the community. We propose the Weighted Community Clustering (WCC), which is a new community metric that takes the triangle instead of the edge as the minimal structural motif indicating the presence of a strong relation in a graph. We theoretically analyse WCC in depth and formally prove, by means of a set of properties, that the maximization of WCC guarantees communities with cohesion and structure. In addition, we propose Scalable Community Detection (SCD), a community detection algorithm based on WCC, which is designed to be fast and scalable on SMP machines, showing experimentally that WCC correctly captures the concept of community in social networks using real datasets. Finally, using ground-truth data, we show that SCD provides better quality than the best disjoint community detection algorithms of the state of the art while performing faster.Polygons as sections of higher-dimensional polytopes
http://hdl.handle.net/2117/86389
Polygons as sections of higher-dimensional polytopes
Padrol Sureda, Arnau; Pfeifle, Julián
We show that every heptagon is a section of a 3-polytope with 6 vertices. This implies that every n-gon with n >= 7 can be obtained as a section of a (2 + [n/7])-dimensional polytope with at most [6n/7] vertices; and provides a geometric proof of the fact that every nonnegative n x rn matrix of rank 3 has nonnegative rank not larger than [6min(n,m)/7]. This result has been independently proved, algebraically, by Shitov (J. Combin. Theory Ser. A 122, 2014).
2016-04-28T14:55:10ZPadrol Sureda, ArnauPfeifle, JuliánWe show that every heptagon is a section of a 3-polytope with 6 vertices. This implies that every n-gon with n >= 7 can be obtained as a section of a (2 + [n/7])-dimensional polytope with at most [6n/7] vertices; and provides a geometric proof of the fact that every nonnegative n x rn matrix of rank 3 has nonnegative rank not larger than [6min(n,m)/7]. This result has been independently proved, algebraically, by Shitov (J. Combin. Theory Ser. A 122, 2014).Lower bounds on the maximum number of non-crossing acyclic graphs
http://hdl.handle.net/2117/86044
Lower bounds on the maximum number of non-crossing acyclic graphs
Huemer, Clemens; Mier Vinué, Anna de
This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration of N points has Omega (12.52(N)) non-crossing spanning trees and Omega (13.61(N)) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Toth (2013). Our analysis relies on the tools of analytic combinatorics, which enable us to count certain families of forests on points in convex position, and to estimate their average number of components. A new upper bound of O(22.12(N)) for the number of non-crossing spanning trees of the double chain is also obtained. (C) 2015 Elsevier Ltd. All rights reserved.
2016-04-21T10:27:51ZHuemer, ClemensMier Vinué, Anna deThis paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration of N points has Omega (12.52(N)) non-crossing spanning trees and Omega (13.61(N)) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Toth (2013). Our analysis relies on the tools of analytic combinatorics, which enable us to count certain families of forests on points in convex position, and to estimate their average number of components. A new upper bound of O(22.12(N)) for the number of non-crossing spanning trees of the double chain is also obtained. (C) 2015 Elsevier Ltd. All rights reserved.Finite automata for Schreier graphs of virtually free groups
http://hdl.handle.net/2117/83851
Finite automata for Schreier graphs of virtually free groups
Silva, Pedro V.; Soler Escrivà, Xaro; Ventura Capell, Enric
The Stallings construction for f.g. subgroups of free groups is generalized by introducing the concept of Stallings section, which allows efficient computation of the core of a Schreier graph based on edge folding. It is proved that the groups that admit Stallings sections are precisely the f.g. virtually free groups, this is proved through a constructive approach based on Bass-Serre theory. Complexity issues and applications are also discussed.
2016-03-04T17:02:48ZSilva, Pedro V.Soler Escrivà, XaroVentura Capell, EnricThe Stallings construction for f.g. subgroups of free groups is generalized by introducing the concept of Stallings section, which allows efficient computation of the core of a Schreier graph based on edge folding. It is proved that the groups that admit Stallings sections are precisely the f.g. virtually free groups, this is proved through a constructive approach based on Bass-Serre theory. Complexity issues and applications are also discussed.On the probability of planarity of a random graph near the critical point
http://hdl.handle.net/2117/83733
On the probability of planarity of a random graph near the critical point
Noy Serrano, Marcos; Ravelomanana, Vlady; Rue, Juanjo
Let G(n, M) be the uniform random graph with n vertices and M edges. Erdos and Renyi (1960) conjectured that the limiting probability; lim(n ->infinity) Pr{G(n, n/2) is planar}; exists and is a constant strictly between 0 and 1. Luczak, Pittel and Wierman (1994) proved this conjecture, and Janson, Luczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact limiting probability of a random graph being planar near the critical point M = n/2. For each lambda, we find an exact analytic expression for; p(lambda) = lim(n ->infinity) Pr {G (n, n/2 (1 + lambda(n-1/3))) is planar}.; In particular, we obtain p(0) approximate to 0.99780. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of G(n, n/2) being series-parallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.
2016-03-02T18:49:21ZNoy Serrano, MarcosRavelomanana, VladyRue, JuanjoLet G(n, M) be the uniform random graph with n vertices and M edges. Erdos and Renyi (1960) conjectured that the limiting probability; lim(n ->infinity) Pr{G(n, n/2) is planar}; exists and is a constant strictly between 0 and 1. Luczak, Pittel and Wierman (1994) proved this conjecture, and Janson, Luczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact limiting probability of a random graph being planar near the critical point M = n/2. For each lambda, we find an exact analytic expression for; p(lambda) = lim(n ->infinity) Pr {G (n, n/2 (1 + lambda(n-1/3))) is planar}.; In particular, we obtain p(0) approximate to 0.99780. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of G(n, n/2) being series-parallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.The conjugacy problem for free-by-cyclic groups
http://hdl.handle.net/2117/79985
The conjugacy problem for free-by-cyclic groups
Martino, Armando; Ventura Capell, Enric
We show that the conjugacy problem is solvable in [finitely
generated free]-by-cyclic groups, by using a result of O. Maslakova
that one can algorithmically find generating sets for the fixed sub-
groups of free group automorphisms, and one of P. Brinkmann that
one can determine whether two cyclic words in a free group are
mapped to each other by some power of a given automorphism. The
algorithm effectively computes a conjugating element, if it exists. We
also solve the power conjugacy problem and give an algorithm to rec-
ognize if two given elements of a finitely generated free group are
Reidemeister equivalent with respect to a given automorphism.
2015-11-26T18:18:08ZMartino, ArmandoVentura Capell, EnricWe show that the conjugacy problem is solvable in [finitely
generated free]-by-cyclic groups, by using a result of O. Maslakova
that one can algorithmically find generating sets for the fixed sub-
groups of free group automorphisms, and one of P. Brinkmann that
one can determine whether two cyclic words in a free group are
mapped to each other by some power of a given automorphism. The
algorithm effectively computes a conjugating element, if it exists. We
also solve the power conjugacy problem and give an algorithm to rec-
ognize if two given elements of a finitely generated free group are
Reidemeister equivalent with respect to a given automorphism.The automorphism group of a free-by-cyclic group in rank 2
http://hdl.handle.net/2117/79983
The automorphism group of a free-by-cyclic group in rank 2
Bogopolski, Oleg; Martino, Armando; Ventura Capell, Enric
Let
¡
be an automorphism of a free group
F
2
of rank
2 and let
M
¡
=
F
2
o
¡
Z
be the corresponding mapping torus of
¡
.
We prove that the group
Out
(
M
¡
) is usually virtually cyclic. More-
over, we classify the cases when this group is Ønite depending on the
conjugacy class of the image of
¡
in
GL
2
(
Z
)
2015-11-26T17:54:44ZBogopolski, OlegMartino, ArmandoVentura Capell, EnricLet
¡
be an automorphism of a free group
F
2
of rank
2 and let
M
¡
=
F
2
o
¡
Z
be the corresponding mapping torus of
¡
.
We prove that the group
Out
(
M
¡
) is usually virtually cyclic. More-
over, we classify the cases when this group is Ønite depending on the
conjugacy class of the image of
¡
in
GL
2
(
Z
)Absolute-type shaft encoding using LFSR sequences with a prescribed length
http://hdl.handle.net/2117/79981
Absolute-type shaft encoding using LFSR sequences with a prescribed length
Fuertes Armengol, José Mª; Balle Pigem, Borja de; Ventura Capell, Enric
Maximal-length binary sequences have existed for a long time. They have many interesting properties, and one of them is that, when taken in blocks of n consecutive positions, they form 2n - 1 different codes in a closed circular sequence. This property can be used to measure absolute angular positions as the circle can be divided into as many parts as different codes can be retrieved. This paper describes how a closed binary sequence with an arbitrary length can be effectively designed with the minimal possible block length using linear feedback shift registers. Such sequences can be used to measure a specified exact number of angular positions using the minimal possible number of sensors that linear methods allow.
2015-11-26T17:41:12ZFuertes Armengol, José MªBalle Pigem, Borja deVentura Capell, EnricMaximal-length binary sequences have existed for a long time. They have many interesting properties, and one of them is that, when taken in blocks of n consecutive positions, they form 2n - 1 different codes in a closed circular sequence. This property can be used to measure absolute angular positions as the circle can be divided into as many parts as different codes can be retrieved. This paper describes how a closed binary sequence with an arbitrary length can be effectively designed with the minimal possible block length using linear feedback shift registers. Such sequences can be used to measure a specified exact number of angular positions using the minimal possible number of sensors that linear methods allow.