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The product graph G m * G p of two given graphs G m and G p was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32-48]. For this kind of graphs we provide bounds for two connectivity parameters ( and , edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs G m ,

"Discrete Mathemetics Top Cited Article 2005-2010"The restricted connectivity κ′(G)κ′(G) of a connected graph G is defined as the minimum cardinality of a vertex-cut over all vertex-cuts X such that no vertex uu has all its neighbors in X; the superconnectivity κ1(G)κ1(G) is defined similarly, this time considering only vertices uu in G-XG-X, hence κ1(G)⩽κ′(G)κ1(G)⩽κ′(G). The minimum edge-degree of G is ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, d(u)d(u) standing for the degree of a vertex uu. In this paper, several sufficient conditions yielding κ1(G)⩾ξ(G)κ1(G)⩾ξ(G) are given, improving a previous related result by Fiol et al. [Short paths and connectivity in graphs and digraphs, Ars Combin. 29B (1990) 17–31] and guaranteeing κ1(G)=κ′(G)=ξ(G)κ1(G)=κ′(G)=ξ(G) under some additional constraints.Peer ReviewedAward-winningPostprint (published version

Abstract. In this note we show a new upperbound for the function ex(n; T K p ), 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 2n+5 3≤ p < n we provide exact values for this function.

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.

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.

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Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209-218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then (g, h) is called the girth pair of the graph. In this paper we prove that a graph with girth pair (g, h) such that g is odd and h g + 3 is even has high (vertex-)connectivity if its diameter is at most h − 3. The edge version of all results is also studied.