COMBGRAF - Combinatòria, Teoria de Grafs i Aplicacions
http://hdl.handle.net/2117/3178
Wed, 28 Jun 2017 12:37:22 GMT2017-06-28T12:37:22ZFrom subKautz digraphs to cyclic Kautz digraphs
http://hdl.handle.net/2117/105173
From subKautz digraphs to cyclic Kautz digraphs
Dalfó Simó, Cristina
Kautz digraphs K(d,l) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz digraphs CK(d,l) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to cyclic Kautz digraphs by introducing the family of subKautz digraphs sK(d,l), from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices of both sK(d,l) and CK(d,l). Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices. Using these parameters, we also prove that sK(d,l) and CK(d,l) are maximally vertex-connected and super-edge-connected. Whereas K(d,l) are optimal with respect to the diameter, we show that sK(d,l) and CK(d,l) are optimal with respect to the mean distance, whose exact values are given for both families when l = 3. Finally, we provide a lower bound on the girth of CK(d,l) and sK(d,l)
Tue, 06 Jun 2017 12:12:17 GMThttp://hdl.handle.net/2117/1051732017-06-06T12:12:17ZDalfó Simó, CristinaKautz digraphs K(d,l) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz digraphs CK(d,l) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to cyclic Kautz digraphs by introducing the family of subKautz digraphs sK(d,l), from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices of both sK(d,l) and CK(d,l). Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices. Using these parameters, we also prove that sK(d,l) and CK(d,l) are maximally vertex-connected and super-edge-connected. Whereas K(d,l) are optimal with respect to the diameter, we show that sK(d,l) and CK(d,l) are optimal with respect to the mean distance, whose exact values are given for both families when l = 3. Finally, we provide a lower bound on the girth of CK(d,l) and sK(d,l)On quotient digraphs and voltage digraphs
http://hdl.handle.net/2117/105171
On quotient digraphs and voltage digraphs
Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel; Miller, Mirka; Ryan, Joe
In this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show their close relationship between voltage digraphs and line digraphs, which are two known approaches to obtain dense digraphs. In the same context, we show the equivalence
between the vertex-splitting and partial line digraph techniques. Then, we give a sufficient condition for a lifted digraph of a base line digraph to be again a line digraph. Some of the results are illustrated with two well-known families of digraphs. Namely, De Bruijn and Kautz digraphs.
Tue, 06 Jun 2017 12:00:49 GMThttp://hdl.handle.net/2117/1051712017-06-06T12:00:49ZDalfó Simó, CristinaFiol Mora, Miquel ÀngelMiller, MirkaRyan, JoeIn this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show their close relationship between voltage digraphs and line digraphs, which are two known approaches to obtain dense digraphs. In the same context, we show the equivalence
between the vertex-splitting and partial line digraph techniques. Then, we give a sufficient condition for a lifted digraph of a base line digraph to be again a line digraph. Some of the results are illustrated with two well-known families of digraphs. Namely, De Bruijn and Kautz digraphs.A new labeling construction from the -product
http://hdl.handle.net/2117/104632
A new labeling construction from the -product
López Masip, Susana Clara; Muntaner Batle, Francesc Antoni; Prabu, M.
The ¿h-product that is referred in the title was introduced in 2008 as a generalization of the Kronecker product of digraphs. Many relations among labelings have been obtained since then, always using as a second factor a family of super edge-magic graphs with equal order and size. In this paper, we introduce a new labeling construction by changing the role of the factors. Using this new construction the range of applications grows up considerably. In particular, we can increase the information about magic sums of cycles and crowns.
Fri, 19 May 2017 07:50:07 GMThttp://hdl.handle.net/2117/1046322017-05-19T07:50:07ZLópez Masip, Susana ClaraMuntaner Batle, Francesc AntoniPrabu, M.The ¿h-product that is referred in the title was introduced in 2008 as a generalization of the Kronecker product of digraphs. Many relations among labelings have been obtained since then, always using as a second factor a family of super edge-magic graphs with equal order and size. In this paper, we introduce a new labeling construction by changing the role of the factors. Using this new construction the range of applications grows up considerably. In particular, we can increase the information about magic sums of cycles and crowns.Locating-dominating partitions in graphs
http://hdl.handle.net/2117/104422
Locating-dominating partitions in graphs
Pelayo Melero, Ignacio Manuel; Hernando Martín, María del Carmen; Mora Giné, Mercè
Let G = (V, E) be a connected graph of order n. Let ¿ = {S1, . . . , Sk} be
a partition of V . Let r(u|¿) denote the vector of distances between a vertex
v ¿ V and the elements of ¿, that is, r(v, ¿) = (d(v, S1), . . . , d(v, Sk)). The
partition ¿ is called a locating partition of G if, for every pair of distinct
vertices u, v ¿ V , r(u, ¿) 6= r(v, ¿). A locating partition ¿ is called metriclocating-dominating partition (an MLD-partition for short) of G if it is also dominating,
Mon, 15 May 2017 12:02:03 GMThttp://hdl.handle.net/2117/1044222017-05-15T12:02:03ZPelayo Melero, Ignacio ManuelHernando Martín, María del CarmenMora Giné, MercèLet G = (V, E) be a connected graph of order n. Let ¿ = {S1, . . . , Sk} be
a partition of V . Let r(u|¿) denote the vector of distances between a vertex
v ¿ V and the elements of ¿, that is, r(v, ¿) = (d(v, S1), . . . , d(v, Sk)). The
partition ¿ is called a locating partition of G if, for every pair of distinct
vertices u, v ¿ V , r(u, ¿) 6= r(v, ¿). A locating partition ¿ is called metriclocating-dominating partition (an MLD-partition for short) of G if it is also dominating,On the spectra of Markov matrices for weighted Sierpinski graphs
http://hdl.handle.net/2117/104412
On the spectra of Markov matrices for weighted Sierpinski graphs
Comellas Padró, Francesc de Paula; Xie, Pinchen; Zhang, Zhongzhi
Relevant information from networked systems can be obtained by analyzing the spectra of matrices associated to their graph representations. In particular, the eigenvalues and eigenvectors of the Markov matrix and related Laplacian and normalized Laplacian matrices allow the study of structural and dynamical aspects of a network, like its synchronizability and random walks properties. In this study we obtain, in a recursive way, the spectra of Markov matrices of a family of rotationally invariant weighted Sierpinski graphs. From them we find analytic expressions for the weighted count of spanning trees and the random target access time for random walks on this family of weighted graphs.
Mon, 15 May 2017 10:27:27 GMThttp://hdl.handle.net/2117/1044122017-05-15T10:27:27ZComellas Padró, Francesc de PaulaXie, PinchenZhang, ZhongzhiRelevant information from networked systems can be obtained by analyzing the spectra of matrices associated to their graph representations. In particular, the eigenvalues and eigenvectors of the Markov matrix and related Laplacian and normalized Laplacian matrices allow the study of structural and dynamical aspects of a network, like its synchronizability and random walks properties. In this study we obtain, in a recursive way, the spectra of Markov matrices of a family of rotationally invariant weighted Sierpinski graphs. From them we find analytic expressions for the weighted count of spanning trees and the random target access time for random walks on this family of weighted graphs.The number and degree distribution of spanning trees in the Tower of Hanoi graph
http://hdl.handle.net/2117/104393
The number and degree distribution of spanning trees in the Tower of Hanoi graph
Zhang, Zhongzhi; Wu, Shunqi; Li, Mingyun; Comellas Padró, Francesc de Paula
The number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph, such as its reliability, communication aspects, synchronization, and so on. However, the practical enumeration of spanning trees and the study of their properties remain a challenge, particularly for large networks. In this paper, we study the number and degree distribution of the spanning trees in the Hanoi graph. We first establish recursion relations between the number of spanning trees and other spanning subgraphs of the Hanoi graph, from which we find an exact analytical expression for the number of spanning trees of the n-disc Hanoi graph. This result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree. Then, we introduce a vertex labeling which allows to find, for each vertex of the graph, its degree distribution among all possible spanning trees.
Mon, 15 May 2017 07:39:12 GMThttp://hdl.handle.net/2117/1043932017-05-15T07:39:12ZZhang, ZhongzhiWu, ShunqiLi, MingyunComellas Padró, Francesc de PaulaThe number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph, such as its reliability, communication aspects, synchronization, and so on. However, the practical enumeration of spanning trees and the study of their properties remain a challenge, particularly for large networks. In this paper, we study the number and degree distribution of the spanning trees in the Hanoi graph. We first establish recursion relations between the number of spanning trees and other spanning subgraphs of the Hanoi graph, from which we find an exact analytical expression for the number of spanning trees of the n-disc Hanoi graph. This result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree. Then, we introduce a vertex labeling which allows to find, for each vertex of the graph, its degree distribution among all possible spanning trees.Iterated line digraphs are asymptotically dense
http://hdl.handle.net/2117/104363
Iterated line digraphs are asymptotically dense
Dalfó Simó, Cristina
We show that the line digraph technique, when iterated, provides dense digraphs, that is, with asymptotically large order for a given diameter (or with small diameter for a given order). This is a well- known result for regular digraphs. In this note we prove that this is also true for non-regular digraphs
Fri, 12 May 2017 11:38:16 GMThttp://hdl.handle.net/2117/1043632017-05-12T11:38:16ZDalfó Simó, CristinaWe show that the line digraph technique, when iterated, provides dense digraphs, that is, with asymptotically large order for a given diameter (or with small diameter for a given order). This is a well- known result for regular digraphs. In this note we prove that this is also true for non-regular digraphsThe normalized Laplacian spectrum of subdivisions of a graph
http://hdl.handle.net/2117/104267
The normalized Laplacian spectrum of subdivisions of a graph
Xie, Pinchen; Zhang, Zhongzhi; Comellas Padró, Francesc de Paula
Determining and analyzing the spectra of graphs is an important and exciting research topic in mathematics science and theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees.
Wed, 10 May 2017 11:31:59 GMThttp://hdl.handle.net/2117/1042672017-05-10T11:31:59ZXie, PinchenZhang, ZhongzhiComellas Padró, Francesc de PaulaDetermining and analyzing the spectra of graphs is an important and exciting research topic in mathematics science and theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees.On perfect and quasiperfect dominations in graphs
http://hdl.handle.net/2117/104244
On perfect and quasiperfect dominations in graphs
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Cáceres, José; Puertas, M. Luz
A subset S ¿ V in a graph G = ( V , E ) is a k -quasiperfect dominating set (for k = 1) if every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k -quasiperfect dominating set in G is denoted by ¿ 1 k ( G ). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n = ¿ 11 ( G ) = ¿ 12 ( G ) = ... = ¿ 1 ¿ ( G ) = ¿ ( G ) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ¿ 12 ( G ) = ¿ ( G ). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ¿ ( G ).
Wed, 10 May 2017 05:56:12 GMThttp://hdl.handle.net/2117/1042442017-05-10T05:56:12ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelCáceres, JoséPuertas, M. LuzA subset S ¿ V in a graph G = ( V , E ) is a k -quasiperfect dominating set (for k = 1) if every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k -quasiperfect dominating set in G is denoted by ¿ 1 k ( G ). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n = ¿ 11 ( G ) = ¿ 12 ( G ) = ... = ¿ 1 ¿ ( G ) = ¿ ( G ) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ¿ 12 ( G ) = ¿ ( G ). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ¿ ( G ).Perfect (super) Edge-Magic Crowns
http://hdl.handle.net/2117/103956
Perfect (super) Edge-Magic Crowns
López Masip, Susana Clara; Muntaner Batle, Francesc Antoni; Prabu, M.
A graph G is called edge-magic if there is a bijective function f from the set of vertices and edges to the set {1,2,…,|V(G)|+|E(G)|} such that the sum f(x)+f(xy)+f(y) for any xy in E(G) is constant. Such a function is called an edge-magic labelling of G and the constant is called the valence. An edge-magic labelling with the extra property that f(V(G))={1,2,…,|V(G)|} is called super edge-magic. A graph is called perfect (super) edge-magic if all theoretical (super) edge-magic valences are possible. In this paper we continue the study of the valences for (super) edge-magic labelings of crowns Cm¿K¯¯¯¯¯n and we prove that the crowns are perfect (super) edge-magic when m=pq where p and q are different odd primes. We also provide a lower bound for the number of different valences of Cm¿K¯¯¯¯¯n, in terms of the prime factors of m.
Wed, 03 May 2017 11:17:49 GMThttp://hdl.handle.net/2117/1039562017-05-03T11:17:49ZLópez Masip, Susana ClaraMuntaner Batle, Francesc AntoniPrabu, M.A graph G is called edge-magic if there is a bijective function f from the set of vertices and edges to the set {1,2,…,|V(G)|+|E(G)|} such that the sum f(x)+f(xy)+f(y) for any xy in E(G) is constant. Such a function is called an edge-magic labelling of G and the constant is called the valence. An edge-magic labelling with the extra property that f(V(G))={1,2,…,|V(G)|} is called super edge-magic. A graph is called perfect (super) edge-magic if all theoretical (super) edge-magic valences are possible. In this paper we continue the study of the valences for (super) edge-magic labelings of crowns Cm¿K¯¯¯¯¯n and we prove that the crowns are perfect (super) edge-magic when m=pq where p and q are different odd primes. We also provide a lower bound for the number of different valences of Cm¿K¯¯¯¯¯n, in terms of the prime factors of m.