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.
A Relation Between Binomial Coefficients and Fibonacci Numbers to the Higher Power
