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

Abstract

In 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 r There exists a one-to-one correspondence between the isomorphism classes of the connected components of the zig-zag product and the isomorphism classes of the corresponding pseudo-replacement graphs (Theorem 4.7). r In the case G 2 C 4 , the connected components of the zig-zag product are isomorphic to double cycle graphs DC n , for some n (Proposition 5.6).r If { n } n≥1 is the sequence of Schreier graphs associated with the action of the Basilica group, then, for each n ≥ 1, the graph n z C 4 is connected and isomorphic to the double cycle graph DC 2 n+1 (Proposition 6.1); the spectral analysis of the graphs n z C 4 is explicitly performed (Theorem 6.5). PRELIMINARIESIn this section, we introduce the replacement and the zig-zag product of two regular graphs. For this, we recall first some basic definitions and properties of regular graphs, and we fix the notation for the rest of the article.Let G = (V, E ) be a finite undirected graph, where V and E denote the vertex set and the edge set of G, respectively. In other words, the elements of the edge set E are unordered Journal of Graph Theory DOI 10.1002/jgt vertex set V 1 × V 2 , that we can identify with the set V 1 × [d 1 ], and whose edges are described by the following rotation map:One can imagine that the vertex set of G 1 r G 2 is partitioned into clouds, which are indexed by the vertices of G 1 , where by definition the v-cloud, for v ∈ V 1 , consists of vertices (v, 1), (v, 2), . . . , (v, d 1 ). Within this construction, the idea is to put a copy of G 2 around each vertex v of G 1 , while keeping edges of both G 1 and G 2 . Every vertex of G 1 r G 2 will be connected to its original neighbors within its cloud (by edges coming from G 2 ), but also to one vertex of a different cloud, according to the rotation map of G 1 . Note that the degree of G 1 r G 2 depends only on the degree of the second factor graph G 2 .Remark 2.2. Notice that the definition of G 1 r G 2 depends on the bi-labeling of G 1 . In general, there may exist two different bi-labelings of G 1 , such that the associated replacement products are nonisomorphic graphs [1, Example 2.3]. Zig-Zag Product of GraphsThe zig-zag product of two graphs was introduced in [25] as a construction which produces, starting from a large graph G 1 and a small graph G 2 , a new graph G 1 z G 2 . T...

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Zeta functions of finite Schreier graphs and their zig-zag products

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