ON DISTANCES IN SIERPIŃSKI GRAPHS: ALMOST-EXTREME VERTICES AND METRIC DIMENSION

Sierpiński graphs $S_p^n$ form an extensively studied family of graphs of fractal nature applicable in topology, mathematics of the Tower of Hanoi, computer science, and elsewhere. An almost-extreme vertex of $S_p^n$ is introduced as a vertex that is either adjacent to an extreme vertex of $S_p^n$ or is incident to an edge between two subgraphs of $S_p^n$ isomorphic to $S_p^{n - 1}$. Explicit formulas are given for the distance in $S_p^n$ between an arbitrary vertex and an almost-extreme vertex. The formulas are applied to compute the total distance of almost-extreme vertices and to obtain the metric dimension of Sierpiński graphs.