Generalized Sierpiński graphs 1

Sierpiński graphs, S(n, k), were defined originally in 1997 by Klavžar and Milutinović. The graph S(1, k) is simply the complete graph Kk and S(n, 3) are the graphs of Tower of Hanoi problem. We generalize the notion of Sierpiński graphs, replacing the complete graph appearing in the case S(1, k) with any graph. The newly introduced notion of generalized Sierpiński graphs can be seen as a criteria to define a graph to be self-similar. We describe the automorphism group of those graphs and compute their distinguishing number. We also study existence of perfect codes in those graphs and give a complete characterization of the existence of perfect codes in the case when the basic graph is a power of a cycle.