8,448 research outputs found

    Metric dimension of dual polar graphs

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    A resolving set for a graph Ξ“\Gamma is a collection of vertices SS, chosen so that for each vertex vv, the list of distances from vv to the members of SS uniquely specifies vv. The metric dimension ΞΌ(Ξ“)\mu(\Gamma) is the smallest size of a resolving set for Ξ“\Gamma. We consider the metric dimension of the dual polar graphs, and show that it is at most the rank over R\mathbb{R} of the incidence matrix of the corresponding polar space. We then compute this rank to give an explicit upper bound on the metric dimension of dual polar graphs.Comment: 8 page

    Metric Dimension of Amalgamation of Graphs

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    A set of vertices SS resolves a graph GG if every vertex is uniquely determined by its vector of distances to the vertices in SS. The metric dimension of GG is the minimum cardinality of a resolving set of GG. Let {G1,G2,…,Gn}\{G_1, G_2, \ldots, G_n\} be a finite collection of graphs and each GiG_i has a fixed vertex v0iv_{0_i} or a fixed edge e0ie_{0_i} called a terminal vertex or edge, respectively. The \emph{vertex-amalgamation} of G1,G2,…,GnG_1, G_2, \ldots, G_n, denoted by Vertexβˆ’Amal{Gi;v0i}Vertex-Amal\{G_i;v_{0_i}\}, is formed by taking all the GiG_i's and identifying their terminal vertices. Similarly, the \emph{edge-amalgamation} of G1,G2,…,GnG_1, G_2, \ldots, G_n, denoted by Edgeβˆ’Amal{Gi;e0i}Edge-Amal\{G_i;e_{0_i}\}, is formed by taking all the GiG_i's and identifying their terminal edges. Here we study the metric dimensions of vertex-amalgamation and edge-amalgamation for finite collection of arbitrary graphs. We give lower and upper bounds for the dimensions, show that the bounds are tight, and construct infinitely many graphs for each possible value between the bounds.Comment: 9 pages, 2 figures, Seventh Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications (CSGT2013), revised version 21 December 201

    The Einstein Relation on Metric Measure Spaces

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    This note is based on F. Burghart's master thesis at Stuttgart university from July 2018, supervised by Prof. Freiberg. We review the Einstein relation, which connects the Hausdorff, local walk and spectral dimensions on a space, in the abstract setting of a metric measure space equipped with a suitable operator. This requires some twists compared to the usual definitions from fractal geometry. The main result establishes the invariance of the three involved notions of fractal dimension under bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more generally, how the transport of the analytic and stochastic structure behind the Einstein relation works. While any homeomorphism suffices for this transport of structure, non-Lipschitz maps distort the Hausdorff and the local walk dimension in different ways. To illustrate this, we take a look at H\"older regular transformations and how they influence the local walk dimension and prove some partial results concerning the Einstein relation on graphs of fractional Brownian motions. We conclude by giving a short list of further questions that may help building a general theory of the Einstein relation.Comment: 28 pages, 3 figure
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