DSpace Collection:
http://hdl.handle.net/2117/3180
2014-08-22T04:04:22ZJacobi matrices and boundary value problems in distance-regular graphs
http://hdl.handle.net/2117/14821
Title: Jacobi matrices and boundary value problems in distance-regular graphs
Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gago Álvarez, Silvia
Abstract: In this work we analyze regular boundary value problems on a distance-regular graph associated with SchrÄodinger operators. These problems include the cases in which the boundary has two or one vertices. Moreover, we obtain the Green matrix for each regular problem. In each case, the Green matrices are given in terms of two families of orthogonal polynomials one of them corresponding with the distance polynomials of the distance-regular graphs.2012-01-26T10:21:55ZA differential approach for bounding the index of graphs under perturbations
http://hdl.handle.net/2117/12078
Title: A differential approach for bounding the index of graphs under perturbations
Authors: Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest2011-03-25T15:57:30ZEdge-distance-regular graphs
http://hdl.handle.net/2117/11794
Title: Edge-distance-regular graphs
Authors: Cámara Vallejo, Marc; Dalfó Simó, Cristina; Fàbrega Canudas, José; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Abstract: Edge-distance-regularity is a concept recently introduced by the authors which is
similar to that of distance-regularity, but now the graph is seen from each of its edges
instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edge-distance-regular when it is distance-regular around each of its edges and with
the same intersection numbers for any edge taken as a root. In this paper we study
this concept, give some of its properties, such as the regularity of Γ, and derive some
characterizations. In particular, it is shown that a graph is edge-distance-regular if and only if its k-incidence matrix is a polynomial of degree k in A multiplied by the
(standard) incidence matrix. Also, the analogue of the spectral excess theorem for
distance-regular graphs is proved, so giving a quasi-spectral characterization of edgedistance-regularity. Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.2011-03-14T08:32:42ZConnected graph searching
http://hdl.handle.net/2117/9116
Title: Connected graph searching
Authors: Barrière Figueroa, Eulalia; Flocchini, Paola; Fomin, Fedor V.; Fraigniaud, Pierre; Nisse, Nicolas; Santoro, Nicola; Thilikos Touloupas, Dimitrios
Abstract: In graph searching game the opponents are a set of searchers and a fugitive in a graph.
The searchers try to capture the fugitive by applying some sequence moves that include
placement, removal, or sliding of a searcher along an edge. The fugitive tries to avoid capture
by moving along unguarded paths. The search number of a graph is the minimum number
of searchers required to guarantee the capture of the fugitive. In this paper, we initiate
the study of this game under the natural restriction of connectivity where we demand that
in each step of the search the locations of the graph that are clean (i.e. non-accessible to
the fugitive) remain connected. We give evidence that many of the standard mathematical
tools used so far in the classic graph searching fail under the connectivity requirement. We
also settle the question on “the price of connectivity” that is how many searchers more
are required for searching a graph when the connectivity demand is imposed. We make
estimations of the price of connectivity on general graphs and we provide tight bounds
for the case of trees. In particular for an n-vertex graph the ratio between the connected
searching number and the non-connected one is O(log n) while for trees this ratio is always
at most 2. We also conjecture that this constant-ratio upper bound for trees holds also for
all graphs. Our combinatorial results imply a complete characterization of connected graph
searching on trees. It is based on a forbidden-graph characterization of the connected search
number. We prove that the connected search game is monotone for trees, i.e. restricting
search strategies to only those where the clean territories increase monotonically does not
require more searchers. A consequence of our results is that the connected search number can
be computed in polynomial time on trees, moreover, we show how to make this algorithm
distributed. Finally, we reveal connections of this parameter to other invariants on trees
such as the Horton-Stralher number.2010-09-27T17:25:19ZStrong product of graphs: Geodetic and hull number and boundary-type sets
http://hdl.handle.net/2117/8413
Title: Strong product of graphs: Geodetic and hull number and boundary-type sets
Authors: Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas González, María Luz2010-07-27T09:58:31ZMean first-passage time for random walks on generalized deterministic recursive trees
http://hdl.handle.net/2117/7385
Title: Mean first-passage time for random walks on generalized deterministic recursive trees
Authors: Comellas Padró, Francesc de Paula; Miralles de la Asunción, Alicia2010-05-26T15:13:42ZGraphs, Friends and Acquaintances
http://hdl.handle.net/2117/7159
Title: Graphs, Friends and Acquaintances
Authors: Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel
Abstract: As is well known, a graph is a mathematical object modeling the
existence of a certain relation between pairs of elements of a given set.
Therefore, it is not surprising that many of the first results concerning
graphs made reference to relationships between people or groups of
people. In this article, we comment on four results of this kind, which
are related to various general theories on graphs and their applications:
the Handshake lemma (related to graph colorings and Boolean
algebra), a lemma on known and unknown people at a cocktail party
(to Ramsey theory), a theorem on friends in common (to distanceregularity
and coding theory), and Hall’s Marriage theorem (to the
theory of networks). These four areas of graph theory, often with
problems which are easy to state but difficult to solve, are extensively
developed and currently give rise to much research work. As examples
of representative problems and results of these areas, which are
discussed in this paper, we may cite the following: the Four Colors
Theorem (4CTC), the Ramsey numbers, problems of the existence of
distance-regular graphs and completely regular codes, and finally the
study of topological proprieties of interconnection networks.2010-05-11T15:41:55ZBounds on the size of super edge-magic graphs depending on the girth
http://hdl.handle.net/2117/7063
Title: Bounds on the size of super edge-magic graphs depending on the girth
Authors: Ichishima, Rikio; Muntaner Batle, Francesc Antoni; Rius Font, Miquel
Abstract: Let G = (V,E) be a graph of order p and size q. It is known that if G is super edge-magic
graph then q 2p−3. Furthermore, if G is super edge-magic and q = 2p−3, then the girth
of G is 3. It is also known that if the girth of G is at least 4 and G is super edge-magic then
q 2p − 5. In this paper we show that there are infinitely many graphs which are super
edge-magic, have girth 5, and q = 2p−5. Therefore the maximum size for super edge-magic
graphs of girth 5 cannot be reduced with respect to the maximum size of super edge-magic
graphs of girth 4.2010-04-22T17:19:00ZOn Almost Distance-Regular Graphs
http://hdl.handle.net/2117/6749
Title: On Almost Distance-Regular Graphs
Authors: Dalfó Simó, Cristina; Van Dam, Edwin; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Gorissen, Bram2010-03-22T13:02:28ZGrafs, amics i coneguts
http://hdl.handle.net/2117/6395
Title: Grafs, amics i coneguts
Authors: Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel
Abstract: Com és ben sabut, un graf no dirigit és un objecte matemàtic que modelitza
l’existència d’una certa relació entre parells d’elements d’un conjunt donat. Aleshores, no
és gaire sorprenent que, al començament, molts dels resultats sobre grafs fessin referència a
relacions entre persones o grups de persones. En aquest article, comentem quatre resultats
d’aquest tipus, els quals estan relacionats amb diverses teories generals de grafs i les seves
aplicacions: el lema de les encaixades de mans (relacionat amb la coloració de grafs i
l’àlgebra booleana), un lema sobre els coneguts i desconeguts en una festa (relacionat
amb la teoria de Ramsey), un lema sobre els amics en comú (relacionat amb la distància-
regularitat i la teoria de codis) i el teorema de les noces de Hall (relacionat amb la
connectivitat de les xarxes).2010-02-16T19:03:45ZCombinatorial vs. algebraic characterizations of pseudo-distance-regularity around a set
http://hdl.handle.net/2117/3016
Title: Combinatorial vs. algebraic characterizations of pseudo-distance-regularity around a set
Authors: Cámara Vallejo, Marc; Fàbrega Canudas, José; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Abstract: Given a simple connected graph $\Gamma$ and a subset of its vertices $C$, the pseudo-distance-regularity around $C$ generalizes, for not necessarily regular graphs, the notion of completely regular code.
Up to now, most of the characterizations of
pseudo-distance-regularity has been derived from a combinatorial definition. In this paper we propose an algebraic (Terwilliger-like) approach to this notion, showing its equivalence with the combinatorial one. This allows us to give new proofs of known
results, and also to obtain new characterizations which do not
depend on the so-called $C$-spectrum of $\Gamma$, but only on the
positive eigenvector of its adjacency matrix. In the way, we also
obtain some results relating the local spectra of a vertex set and
its antipodal. As a consequence of our study, we obtain a new characterization of a completely regular code $C$, in terms of the
number of walks in $\Gamma$ with an endvertex in $C$.2009-06-22T16:36:20Z4-labelings and grid embeddings of plane quadrangulations
http://hdl.handle.net/2117/3013
Title: 4-labelings and grid embeddings of plane quadrangulations
Authors: Barrière Figueroa, Eulalia; Huemer, Clemens
Abstract: We show that each quadrangulation on $n$ vertices has a closed rectangle of influence drawing on the $(n-2) \times (n-2)$ grid.
Further, we present a simple algorithm to obtain a straight-line drawing of a quadrangulation on the
$\Big\lceil\frac{n}{2}\Big\rceil \times \Big\lceil\frac{3n}{4}\Big\rceil$ grid.
This is not optimal but has the advantage over other existing algorithms that it is not needed to add edges to
the quadrangulation to make it $4$-connected.
The algorithm is based on angle labeling and simple face counting in regions analogous to Schnyder's grid embedding for triangulation.
This extends previous results on book embeddings for quadrangulations from Felsner, Huemer, Kappes, and Orden (2008).
Our approach also yields a representation of a quadrangulation as a pair of rectangulations with a curious property.2009-06-09T12:59:27ZNotes on betweenness centrality of a graph
http://hdl.handle.net/2117/2891
Title: Notes on betweenness centrality of a graph
Authors: Gago Álvarez, Silvia; Hurajová, Jana; Madaras, Tomas
Abstract: The betweenness centrality of a vertex of a graph is the portion of
shortest paths between all pairs of vertices passing through that
vertex. We study selected general properties of this invariant and
its relations to distance parameters (diameter, mean distance);
also, there are studied properties of graphs whose vertices have the
same value of betweenness centrality.2009-05-20T16:21:53ZOn golden spectral graphs
http://hdl.handle.net/2117/2818
Title: On golden spectral graphs
Authors: Estrada Roger, Ernesto; Gago Álvarez, Silvia
Abstract: The concept of golden spectral graphs is introduced and some of their general
properties reported. Golden spectral graphs are those having a golden proportion for the spectral ratios defined on the basis of the spectral gap, spectral spread and the difference between the second largest and the smallest eigenvalue of the adjacency matrix. They are good expanders and display excellent synchronizability. Here we report some new construction methods as well as several of their topological parameters.2009-03-26T17:19:07Z