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On the Number of Shortest Weighted Paths in a Triangular Grid

Author

Listed:
  • Benedek Nagy

    (Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, North Cyprus, via Mersin 10, Famagusta 99450, Turkey)

  • Bashar Khassawneh

    (Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, North Cyprus, via Mersin 10, Famagusta 99450, Turkey)

Abstract

Counting the number of shortest paths in various graphs is an important and interesting combinatorial problem, especially in weighted graphs with various applications. We consider a specific infinite graph here, namely the honeycomb grid. Changing to its dual, the triangular grid, paths between triangle pixels (we abbreviate this term to trixels) are counted. The number of shortest weighted paths between any two trixels of the triangular grid is discussed. For each trixel, there are three different types of neighbor trixels, 1-, 2- and 3-neighbours, depending the Euclidean distance of their midpoints. When considering weighted distances, the positive values α , β and γ are assigned to the ‘steps’ to various neighbors. We gave formulae for the number of shortest weighted paths between any two trixels in various cases by the respective weight values. The results are nicely connected to various numbers well-known in combinatorics, e.g., to binomial coefficients and Fibonacci numbers.

Suggested Citation

  • Benedek Nagy & Bashar Khassawneh, 2020. "On the Number of Shortest Weighted Paths in a Triangular Grid," Mathematics, MDPI, vol. 8(1), pages 1-16, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:118-:d:308069
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    References listed on IDEAS

    as
    1. Nagy, B., 2002. "Metrics based on neighbourhood sequences in triangular grids," Pure Mathematics and Applications, Department of Mathematics, Corvinus University of Budapest, vol. 13(1-2), pages 259-274.
    2. Gergely Kovács & Benedek Nagy & Béla Vizvári, 2019. "Chamfer distances on the isometric grid: a structural description of minimal distances based on linear programming approach," Journal of Combinatorial Optimization, Springer, vol. 38(3), pages 867-886, October.
    Full references (including those not matched with items on IDEAS)

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