Precious Metal Sequences and
Sierpinski-Type Graphs Andreas M. Hinz¹
Department of Mathematics
Ludwig-Maximilians-Universität München
Theresienstraße 39
80333 Munich
Germany
hinz@math.lmu.de

Paul K. Stockmeyer
Department of Computer Science
The College of William & Mary
P.O. Box 8795
Williamsburg, VA 23187-8795
USA
pkstockmeyer@cox.net

in

Abstract:

Sierpinski graphs $S_p^n$ and Sierpinski triangle graphs $\widehat{S}_p^n$ form two-parametric families of connected simple graphs which are related, for $p=3$, to the Tower of Hanoi with $n$ discs and for $n\rightarrow \infty$ to the Sierpinski triangle fractal. The vertices of minimal degree play a special role as extreme vertices in $S_p^n$ and primitive vertices in $\widehat{S}_p^n$. The key concept of this note is that of an $m$-key vertex whose distance to one of the extreme or primitive vertices, respectively, is $m$ times the distance to another one. The number of such vertices and the distances occurring lead to integer sequences with respect to parameter $n$ like, e.g., the Fibonacci sequence (golden) for $p=3$ and the Pell sequence (silver) for $p=4$. The elements of most of these sequences form self-generating sets. We discuss the cases $m=1,2,3,4$ in detail.