Connectedness and Isomorphism Properties of the Zig‐Zag Product of Graphs

Abstract

AbstractIn this article, we investigate the connectedness and the isomorphism problems for zig‐zag products of two graphs. A sufficient condition for the zig‐zag product of two graphs to be connected is provided, reducing to the study of the connectedness property of a new graph which depends only on the second factor of the graph product. We show that, when the second factor is a cycle graph, the study of the isomorphism problem for the zig‐zag product is equivalent to the study of the same problem for the associated pseudo‐replacement graph. The latter is defined in a natural way, by a construction generalizing the classical replacement product, and its degree is smaller than the degree of the zig‐zag product graph. Two particular classes of products are studied in detail: the zig‐zag product of a complete graph with a cycle graph, and the zig‐zag product of a 4‐regular graph with the cycle graph of length 4. Furthermore, an example coming from the theory of Schreier graphs associated with the action of self‐similar groups is also considered: the graph products are completely determined and their spectral analysis is developed.