http://hdl.handle.net/2117/3180 2013-05-20T06:51:18Z 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. 2012-01-26T10:21:55Z 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, Ernest 2011-03-25T15:57:30Z 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. 2011-03-14T08:32:42Z 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. 2010-09-27T17:25:19Z 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 Luz 2010-07-27T09:58:31Z 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, Alicia 2010-05-26T15:13:42Z 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. 2010-05-11T15:41:55Z 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. 2010-04-22T17:19:00Z 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, Bram 2010-03-22T13:02:28Z 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). 2010-02-16T19:03:45Z 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 $C$. 2009-06-22T16:36:20Z 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. 2009-06-09T12:59:27Z 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. 2009-05-20T16:21:53Z 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