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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The divisible sandpile model is a growth model on graphs that was introduced by Levine and Peres [LP09] as a tool to study internal diffusion limited aggregation. In this work we investigate the shape of the divisible sandpile model on the graphical Sierpinski gasket SG. We show that the shape is a ball in the graph metric of SG. Moreover we give an exact representation of the odometer function of the divisible sandpile.2010 Mathematics Subject Classification. 60G50, 60J10.

We introduce a family of stochastic processes on the integers, depending on a parameter p∈[0,1] and interpolating between the deterministic rotor walk (p=0) and the simple random walk (p=1/2). This p‐rotor walk is not a Markov chain but it has a local Markov property: for each x∈ℤ the sequence of successive exits from x is a Markov chain. The main result of this paper identifies the scaling limit of the p‐rotor walk with two‐sided i.i.d. initial rotors. The limiting process takes the form 1−ppX(t), where X is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation X ( t ) = B ( t ) + a sup ⁡ s ≤ t X ( s ) + b inf ⁡ s ≤ t X ( s ) for all t∈[0,∞). Here B(t) is a standard Brownian motion and a,b<1 are constants depending on the marginals of the initial rotors on ℕ and −ℕ respectively. Chaumont and Doney have shown that Equation 1 has a pathwise unique solution X(t), and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, lim⁡sup⁡X(t)=+∞ and lim⁡inf⁡X(t)=−∞. This last result, together with the main result of this paper, implies that the p‐rotor walk is recurrent for any two‐sided i.i.d. initial rotors and any 0

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