On Some Metric Properties of the Sierpinski Graphs S(n, k)

Sierpinski graphs S(n, k), n, k is an element of N, can be interpreted as graphs of a variant of the Tower of Hanoi with k >= 3 pegs and n >= 1 discs. In particular, it has been proved that for k = 3 the graphs S(n, 3) are isomorphic to the Hanoi graphs H(3)(n). In this paper the chromatic number, the diameter, the eccentricity of a vertex, the radius and the centre of S(n, k) will be determined. Moreover, an important invariant and a number-theoretical characterization of S(n, k) will be derived. By means of these results the complexity of Problem 1, that is the complexity to get from an arbitrary vertex v is an element of S(n, k) to the nearest and to the most distant extreme vertex, will be given. For the Hanoi graphs H(3)(n) some of these results are new.