An advance in infinite graph models for the analysis of transportation networks

This paper extends to infinite graphs the most general extremal issues, which are problems of determining the maximum number of edges of a graph not containing a given subgraph. It also relates the new results with the corresponding situations for the finite case. In particular, concepts from ‘finite’ graph theory, like the average degree and the extremal number, are generalized and computed for some specific cases. Finally, some applications of infinite graphs to the transportation of dangerous goods are presented; they involve the analysis of networks and percolation thresholds.Unión Europea FEDER G-GI3003/IDI

The Size of a Graph Without Topological Complete Subgraphs

In this note we show a new upperbound for the function ex(n;TKp), i.e., the maximum number of edges of a graph of order n not containing a subgraph homeomorphic to the complete graph of order p. Further, for ${\left \lceil \frac{2n+5}{3}\right \rceil}\leq p < n$ we provide exact values for this function

Extremal Graphs without Topological Complete Subgraphs

The exact values of the function $ex(n;TK_{p})$ are known for ${\lceil \frac{2n+5}{3}\rceil}\leq p < n$ (see [Cera, Diánez, and Márquez, SIAM J. Discrete Math., 13 (2000), pp. 295--301]), where $ex(n;TK_p)$ is the maximum number of edges of a graph of order n not containing a subgraph homeomorphic to the complete graph of order $p.$ In this paper, for ${\lceil \frac{2n+6}{3} \rceil}\leq p < n - 3,$ we characterize the family of extremal graphs $EX(n;TK_{p}),$ i.e., the family of graphs with n vertices and $ex(n;TK_{p})$ edges not containing a subgraph homeomorphic to the complete graph of order $p.

Superconnectivity of Networks Modeled by the Strong Product of Graphs

Maximal connectivity and superconnectivity in a network are two important features of its reliability. In this paper, using graph terminology, we first give a lower bound for the vertex connectivity of the strong product of two networks and then we prove that the resulting structure is more reliable than its generators. Namely, sufficient conditions for a strong product of two networks to be maximally connected and superconnected are given.Ministerio de Economía y Competitividad MTM2014-60127-

On the Ramsey numbers for stars versus complete graphs

For graphs G1, . . . , Gs, the multicolor Ramsey number R(G1, . . . , Gs) is the smallest integer r such that if we give any edge col-oring of the complete graph on r vertices with s colors then there exists a monochromatic copy of Gi colored with color i, for some 1 ≤ i ≤ s. In this work the multicolor Ramsey number R(Kp1 , . . . , Kpm , K1,q1 , . . . , K1,qn ) is determined for any set of com-plete graphs and stars in terms of R(Kp1 , . . . , Kpm )Ministerio de Educación y Ciencia MTM2008-06620-C03-02Junta de Andalucía P06-FQM-0164

Avances en Matemática Discreta en Andalucía. V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 2007

V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 200

El tamaño de un grafo sin subgrafos homeomorfos a un grafo completo

"Desde el origen de la Teoría de Grafos Extremales, uno de los problemas más generales que pueden plantearse en este campo, es estudiar los grafos de manera que podamos encontrar condiciones para que contengan o no a un subgrafo dado. Es en este sentido donde podemos encuadrar los objetivos de esta Tesis.Concretamente, nos va interesar el estudio de la función ex (n; TKp), es decir, el número máximo de aristas de un grafo de orden n para que no contenga a un subgrafo homeomorfo al grafo completo de orden p. A su vez, como en todo problema extremal, resulta interesante caracterizar los grafos maximales para la propiedad anterior, esto es lo que se conoce como familia de grafos extremales. Una pequeña variación del problema anterior nos conduce al análisis paralelo de la función ex (n; TK-p).Por otra parte, el estudio de problemas extremales, como los anteriores descritos, cuando el orden de los grafos estudiados es suficientemente grande, conduce de forma natural a plantearse el problema para grafos infinitos. Claro está, que para grafos infinitos, carece de sentido estudiar el número de aristas frente al número de vértices por ser, en general, ambos infinitos. Intentamos dar solución a este problema introduciendo el concepto de valencia media de un grafo infinito como límite de las valencias medias de una sucesión creciente de grafos finitos que lo recubren. Esto, nos permite abordar el problema extremal, relacionado con la contención de subgrafos homeomorfos a un grafo completo, para grafos infinitos en función de la valencia media, así como, establecer las relaciones con el correspondiente problema para el caso finito.

A network approach to analyze neuronal lineage and layer innervation in the Drosophila optic lobes

The optic lobes of the fruit fly Drosophila melanogaster form a highly wired neural network composed of roughly 130.000 neurons of more than 80 different types. How neuronal diversity arises from very few cell progenitors is a central question in developmental neurobiology. We use the optic lobe of the fruit fly as a paradigm to understand how neuroblasts, the neural stem cells, generate multiple neuron types. Although the development of the fly brain has been the subject of extensive research, very little is known about the lineage relationships of the cell types forming the adult optic lobes. Here we perform a large-scale lineage bioinformatics analysis using the graph theory. We generated a large collection of cell clones that genetically label the progeny of neuroblasts and built a database to draw graphs showing the lineage relationships between cell types. By establishing biological criteria that measures the strength of the neuronal relationships and applying community detection tools we have identified eight clusters of neurons. Each cluster contains different cell types that we pose are the product of eight distinct classes of neuroblasts. Three of these clusters match the available lineage data, supporting the predictive value of the analysis. Finally, we show that the neuronal progeny of a neuroblast do not have preferential innervation patterns, but instead become part of different layers and neuropils. Here we establish a new methodology that helps understanding the logic of Drosophila brain development and can be applied to the more complex vertebrate brains.Ministerio de Economía y Competitividad FIS2014-60843-PMinisterio de Economía y Competitividad MTM2014-61312-EX

Locating a Central Hunter on the Plane

Protection, surveillance or other types of coverage services of mobile points call for different, asymmetric distance measures than the traditional Euclidean, rectangular or other norms used for fixed points. In this paper, the destinations are mobile points (prey) moving at fixed speeds and directions and the facility (hunter) can capture them using one of two possible strategies: either it is smart, predicting the prey’s movement in order to minimize the time needed to capture it, or it is dumb, following a pursuit curve, by moving at any moment in the direction of the prey. In either case, the hunter location in a plane is sought in order to minimize the maximum time of capture of any prey. An efficient solution algorithm is developed that uses the particular geometry that both versions of this problem possess. In the case of unpre-dictable movement of prey, a worst-case type solution is proposed, which reduces to the well-known weighted Euclidean minimax location problem.Ministerio de Ciencia y Tecnología BFM2003-04062/MATEMinisterio de Educación y Ciencia MTM2006-1505