COMBGRAF - Combinatòria, Teoria de Grafs i Aplicacions
http://hdl.handle.net/2117/3178
Fri, 28 Oct 2016 12:23:38 GMT2016-10-28T12:23:38ZSymmetry breaking in tournaments
http://hdl.handle.net/2117/91158
Symmetry breaking in tournaments
Lozano Bojados, Antoni
We provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices S is a determining set for a tournament T if every nontrivial automorphism of T moves at least one vertex of S, while S is a resolving set for T if every two distinct vertices in T have different distances to some vertex in S. We show that the minimum size of a determining set for an order n tournament (its determining number) is bounded by n/3, while the minimum size of a resolving set for an order n strong tournament (its metric dimension) is bounded by n/2. Both bounds are optimal.
Thu, 27 Oct 2016 11:29:19 GMThttp://hdl.handle.net/2117/911582016-10-27T11:29:19ZLozano Bojados, AntoniWe provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices S is a determining set for a tournament T if every nontrivial automorphism of T moves at least one vertex of S, while S is a resolving set for T if every two distinct vertices in T have different distances to some vertex in S. We show that the minimum size of a determining set for an order n tournament (its determining number) is bounded by n/3, while the minimum size of a resolving set for an order n strong tournament (its metric dimension) is bounded by n/2. Both bounds are optimal.Distance mean-regular graphs
http://hdl.handle.net/2117/91052
Distance mean-regular graphs
Fiol Mora, Miquel Àngel; Diego, Victor
We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let G be a graph with vertex set V , diameter D, adjacency matrix A, and adjacency algebra A. Then, G is distance mean-regular when, for a given u ¿ V , the averages of the intersection numbers p h ij (u, v) = |Gi(u) n Gj (v)| (number of vertices at distance i from u and distance j from v) computed over all vertices v at a given distance h ¿ {0, 1, . . . , D} from u, do not depend on u. In this work we study some properties and characterizations of these graphs. For instance, it is shown that a distance mean-regular graph is always distance degree-regular, and we give a condition for the converse to be also true. Some algebraic and spectral properties of distance mean-regular graphs are also investigated. We show that, for distance mean regular-graphs, the role of the distance matrices of distance-regular graphs is played for the so-called distance mean-regular matrices. These matrices are computed from a sequence of orthogonal polynomials evaluated at the adjacency matrix of G and, hence, they generate a subalgebra of A. Some other algebras associated to distance mean-regular graphs are also characterized.
Tue, 25 Oct 2016 09:46:43 GMThttp://hdl.handle.net/2117/910522016-10-25T09:46:43ZFiol Mora, Miquel ÀngelDiego, VictorWe introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let G be a graph with vertex set V , diameter D, adjacency matrix A, and adjacency algebra A. Then, G is distance mean-regular when, for a given u ¿ V , the averages of the intersection numbers p h ij (u, v) = |Gi(u) n Gj (v)| (number of vertices at distance i from u and distance j from v) computed over all vertices v at a given distance h ¿ {0, 1, . . . , D} from u, do not depend on u. In this work we study some properties and characterizations of these graphs. For instance, it is shown that a distance mean-regular graph is always distance degree-regular, and we give a condition for the converse to be also true. Some algebraic and spectral properties of distance mean-regular graphs are also investigated. We show that, for distance mean regular-graphs, the role of the distance matrices of distance-regular graphs is played for the so-called distance mean-regular matrices. These matrices are computed from a sequence of orthogonal polynomials evaluated at the adjacency matrix of G and, hence, they generate a subalgebra of A. Some other algebras associated to distance mean-regular graphs are also characterized.José Gómez Martí 1968-2014: in memoriam
http://hdl.handle.net/2117/89974
José Gómez Martí 1968-2014: in memoriam
Fiol Mora, Miquel Àngel
Fri, 16 Sep 2016 10:33:15 GMThttp://hdl.handle.net/2117/899742016-09-16T10:33:15ZFiol Mora, Miquel ÀngelA note on the order of iterated line digraphs
http://hdl.handle.net/2117/89923
A note on the order of iterated line digraphs
Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel
Given a digraph G, we propose a new method to find the recurrence equation for the number of vertices n_k of the k-iterated line digraph L_k(G), for k>= 0, where L_0(G) = G. We obtain this result by using the minimal
polynomial of a quotient digraph pi(G) of G. We show some examples of this method applied to the so-called cyclic Kautz, the unicyclic, and the acyclic digraphs. In the first case, our method gives the enumeration of the ternary length-2 squarefree words of any length.
Wed, 14 Sep 2016 12:40:26 GMThttp://hdl.handle.net/2117/899232016-09-14T12:40:26ZDalfó Simó, CristinaFiol Mora, Miquel ÀngelGiven a digraph G, we propose a new method to find the recurrence equation for the number of vertices n_k of the k-iterated line digraph L_k(G), for k>= 0, where L_0(G) = G. We obtain this result by using the minimal
polynomial of a quotient digraph pi(G) of G. We show some examples of this method applied to the so-called cyclic Kautz, the unicyclic, and the acyclic digraphs. In the first case, our method gives the enumeration of the ternary length-2 squarefree words of any length.Deterministic hierarchical networks
http://hdl.handle.net/2117/89918
Deterministic hierarchical networks
Barrière Figueroa, Eulalia; Comellas Padró, Francesc de Paula; Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel
It has been shown that many networks associated with complex systems are
small-world (they have both a large local clustering coefficient and a small
diameter) and also scale-free (the degrees are distributed according to a power law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems that are modeled. Most of the studies for complex networks are based on stochastic methods. However, a deterministic method, with an exact determination of the main relevant parameters of the networks, has proven useful. Indeed, this approach complements and enhances
the probabilistic and simulation techniques and, therefore, it provides a better understanding of the modeled systems. In this paper we find the radius, diameter, clustering coefficient and degree distribution of a generic family of deterministic hierarchical small-world scale-free networks that has been considered for modeling real-life complex systems.
Wed, 14 Sep 2016 12:16:25 GMThttp://hdl.handle.net/2117/899182016-09-14T12:16:25ZBarrière Figueroa, EulaliaComellas Padró, Francesc de PaulaDalfó Simó, CristinaFiol Mora, Miquel ÀngelIt has been shown that many networks associated with complex systems are
small-world (they have both a large local clustering coefficient and a small
diameter) and also scale-free (the degrees are distributed according to a power law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems that are modeled. Most of the studies for complex networks are based on stochastic methods. However, a deterministic method, with an exact determination of the main relevant parameters of the networks, has proven useful. Indeed, this approach complements and enhances
the probabilistic and simulation techniques and, therefore, it provides a better understanding of the modeled systems. In this paper we find the radius, diameter, clustering coefficient and degree distribution of a generic family of deterministic hierarchical small-world scale-free networks that has been considered for modeling real-life complex systems.The degree/diameter problem in maximal planar bipartite graphs
http://hdl.handle.net/2117/89907
The degree/diameter problem in maximal planar bipartite graphs
Dalfó Simó, Cristina; Huemer, Clemens; Salas Piñon, Julián
The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases.
Wed, 14 Sep 2016 10:33:44 GMThttp://hdl.handle.net/2117/899072016-09-14T10:33:44ZDalfó Simó, CristinaHuemer, ClemensSalas Piñon, JuliánThe (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases.Cospectral digraphs from locally line digraphs
http://hdl.handle.net/2117/89900
Cospectral digraphs from locally line digraphs
Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel
A digraph Gamma = (V, E) is a line digraph when every pair of vertices u, v is an element of V have either equal or disjoint in -neighborhoods. When this condition only applies for vertices in a given subset (with at least two elements), we say that Gamma is a locally line digraph. In this paper we give a new method to obtain a digraph Gamma' cospectral with a given locally line digraph Gamma with diameter D, where the diameter D' of Gamma' is in the interval [D - 1, D + 1]. In particular, when the method is applied to De Bruijn or Kautz digraphs, we obtain cospectral digraphs with the same algebraic properties that characterize the formers. (C) 2016 Elsevier Inc. All rights reserved.
Wed, 14 Sep 2016 09:57:42 GMThttp://hdl.handle.net/2117/899002016-09-14T09:57:42ZDalfó Simó, CristinaFiol Mora, Miquel ÀngelA digraph Gamma = (V, E) is a line digraph when every pair of vertices u, v is an element of V have either equal or disjoint in -neighborhoods. When this condition only applies for vertices in a given subset (with at least two elements), we say that Gamma is a locally line digraph. In this paper we give a new method to obtain a digraph Gamma' cospectral with a given locally line digraph Gamma with diameter D, where the diameter D' of Gamma' is in the interval [D - 1, D + 1]. In particular, when the method is applied to De Bruijn or Kautz digraphs, we obtain cospectral digraphs with the same algebraic properties that characterize the formers. (C) 2016 Elsevier Inc. All rights reserved.On the Partition Dimension and the Twin Number of a Graph
http://hdl.handle.net/2117/87267
On the Partition Dimension and the Twin Number of a Graph
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
A partition of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of . The partition dimension of G is the minimum cardinality of a locating partition of G . A pair of vertices u;v of a graph G are called twins if they have exactly the same set of neighbors other than u and v . A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G . In this paper we undertake the study of the partition dimension of a graph by also considering its twin number. This approach allows us to obtain the set of connected graphs of order n 9 having partition dimension n
Tue, 24 May 2016 10:33:18 GMThttp://hdl.handle.net/2117/872672016-05-24T10:33:18ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelA partition of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of . The partition dimension of G is the minimum cardinality of a locating partition of G . A pair of vertices u;v of a graph G are called twins if they have exactly the same set of neighbors other than u and v . A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G . In this paper we undertake the study of the partition dimension of a graph by also considering its twin number. This approach allows us to obtain the set of connected graphs of order n 9 having partition dimension nPerfect and quasiperfect domination in trees
http://hdl.handle.net/2117/86561
Perfect and quasiperfect domination in trees
Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, Maria Luz
A k quasip erfect dominating set of a connected graph G is a vertex subset S such that every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k-quasip erfect dominating set in G is denoted by 1 k ( G ) . These graph parameters were rst intro duced by Chellali et al. (2013) as a generalization of b oth the p erfect domination numb er 11 ( G ) and the domination numb er ( G ) . The study of the so-called quasip erfect domination chain 11 ( G ) 12 ( G ) 1 ( G ) = ( G ) enable us to analyze how far minimum dominating sets are from b eing p erfect. In this pap er, we provide, for any tree T and any p ositive integer k , a tight upp er b ound of 1 k ( T ) . We also prove that there are trees satisfying all p ossible equalities and inequalities in this chain. Finally a linear algorithm for computing 1 k ( T ) in any tree T is presente
Wed, 04 May 2016 10:50:08 GMThttp://hdl.handle.net/2117/865612016-05-04T10:50:08ZCáceres, JoséHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelPuertas, Maria LuzA k quasip erfect dominating set of a connected graph G is a vertex subset S such that every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k-quasip erfect dominating set in G is denoted by 1 k ( G ) . These graph parameters were rst intro duced by Chellali et al. (2013) as a generalization of b oth the p erfect domination numb er 11 ( G ) and the domination numb er ( G ) . The study of the so-called quasip erfect domination chain 11 ( G ) 12 ( G ) 1 ( G ) = ( G ) enable us to analyze how far minimum dominating sets are from b eing p erfect. In this pap er, we provide, for any tree T and any p ositive integer k , a tight upp er b ound of 1 k ( T ) . We also prove that there are trees satisfying all p ossible equalities and inequalities in this chain. Finally a linear algorithm for computing 1 k ( T ) in any tree T is presenteCounting configuration–free sets in groups
http://hdl.handle.net/2117/86058
Counting configuration–free sets in groups
Rué Perna, Juan José; Serra Albó, Oriol; Vena Cros, Lluís
We present a unified framework to asymptotically count the number of sets, with a given cardinality, free of certain configurations. This is done by combining the hypergraph containers methodology joint with arithmetic removal lemmas. Several applications involving linear configurations are described, as well as some applications in the random sparse setting.
Thu, 21 Apr 2016 11:49:48 GMThttp://hdl.handle.net/2117/860582016-04-21T11:49:48ZRué Perna, Juan JoséSerra Albó, OriolVena Cros, LluísWe present a unified framework to asymptotically count the number of sets, with a given cardinality, free of certain configurations. This is done by combining the hypergraph containers methodology joint with arithmetic removal lemmas. Several applications involving linear configurations are described, as well as some applications in the random sparse setting.