Connectedness and isomorphism properties of the zig-zag product of graphs
In this paper 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 -regular graph with the cycle graph of length . 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.
Key words and phrases:Zig-zag product, replacement product, parity block, parity block decomposition, connected component, pseudo-replacement, double cycle graph.
Mathematics Subject Classification (2010): 05C60, 05C76, 05C78.
The fruitful idea of constructing new graphs starting from smaller factor graphs is very popular in Mathematics and it has been largely studied and developed in the literature for its theoretical interest, as well as for its numerous applications in several branches like Combinatorics, Probability, Theoretical Computer Science, Statistical Mechanics.
This paper is devoted to the study of the connectedness and isomorphism properties of zig-zag products of graphs. This combinatorial construction, which applies to regular graphs, was introduced in  by O. Reingold, S. Vadhan and A. Wigderson, in order to provide new sequences of constant degree expanders of arbitrary size. Informally, a graph is expander if it is simultaneously sparse, i.e., it has relatively few edges, and highly connected. What is mostly fascinating about expander graphs, is the fact that the expansion property can be described from several points of view - combinatorial, algebraic and probabilistic. Expander graphs have many interesting applications in different areas of Computer Science, such as design and analysis of communication networks and error correcting codes, as well as in many computational problems, by playing a crucial role also in Statistical Physics, Computational Group Theory, and Optimization [20, 22].
The zig-zag product is strictly related to a simpler construction, called replacement product of graphs. The replacement and the zig-zag product play an important role in Geometric Group Theory, since it turns out that, when applied to Cayley graphs of two finite groups, they provide the Cayley graph of the semidirect product of these groups . Further results about the relationship between graph products and group operations are given in .
The structure of the paper is as follows. In Section 2 we recall the definition and the basic properties of the replacement and zig-zag product of graphs. In Section 3, we attack the connectedness problem for zig-zag products of regular graphs. In Section 4, we focus our attention on the classification of the isomorphism classes of zig-zag products in the case where the second factor graph is a cycle graph of even length. In this context, we prove that the connected components of the zig-zag product are in one-to-one correspondence with the so-called parity blocks, introduced in Subsection 4.1. These are subgraphs of the first factor of the zig-zag product, considered together with the bi-labelling of its edges. The isomorphism problem is treated by associating with any parity block (and so with any connected component of the zig-zag product) a new simpler graph, that we call pseudo-replacement graph. The pseudo-replacement graph contains, in general, less vertices and edges, and has a smaller degree than the corresponding connected component of the zig-zag product. Nevertheless, it completely encodes the isomorphism properties of each connected component (see Subsection 4.2). In the case where the cycle graph has length , we show that the structure of the zig-zag product is very regular: it consists of highly symmetric graphs that we call double cycle graphs (see Subsection 5.2). In particular, this implies that the zig-zag product of graphs is not injective, as two non isomorphic graphs can produce isomorphic zig-zag products. In this setting, we are also able to perform a complete spectral analysis, by using the fact that the adjacency matrices of the double cycle graphs are circulant.
Interesting sequences of increasing regular graphs can be obtained by considering the Schreier graphs of the action of groups generated by finite automata. In Section 6, we describe an application of the zig-zag product to this setting. The class of automata groups became very popular after the introduction of the Grigorchuk group, that was the first example of a group with intermediate growth (see  for the definition and further references). Surprising deep connections between groups generated by automata, complex dynamics, fractal geometry have been discovered, and they constitute a very exciting topic of investigation in modern mathematics [23, 24]. In particular, sequences of finite Schreier graphs represent a discrete approximation of fractal limit objects associated with such groups. This point of view can also be exploited in the study of models coming from Statistical Mechanics [8, 9, 4, 15].
The main results achieved in the current paper can be summarized as follows.
A sufficient condition for the connectedness of the zig-zag product is given in terms of the connectedness of a new graph , called the neighborhood graph of . The construction of depends only on the structure of and the number of vertices equals the number of vertices of (Theorem 3.1).
There exists a one-to-one correspondence between the parity blocks in the parity block decomposition of , and the connected components of the graph (Theorem 4.1).
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 4.2).
In the case , the connected components of the zig-zag product are isomorphic to double cycle graphs , for some (Proposition 5.3).
In 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 paper.
Let be a finite undirected graph, where and denote the vertex set and the edge set of , respectively. In other words, the elements of the edge set are unordered pairs of type , with . If , we say that the vertices and are adjacent in , and we use the notation . We will also say that the edge joins and . Loops and multi-edges are also allowed. A path in is a sequence of vertices of such that . The graph is connected if, for every , there exists a path in such that and . The degree of a vertex of is defined as . We assume that a loop at the vertex counts twice in the degree of . We say that is a regular graph of degree , or a -regular graph, if for every .
Let and denote by the adjacency matrix of , that is, the square matrix of size indexed by , whose entry equals the number of edges joining and . Note that if there are loops at the vertex . As the graph is undirected, is a symmetric matrix, so that it admits real eigenvalues . One has : in particular, the -regularity condition can be rewritten as , for each . For a -regular graph , the normalized adjacency matrix is defined as . It is known [5, 11] that, if is a -regular graph, with , and is its adjacency matrix, then is an eigenvalue of . Its multiplicity as an eigenvalue of equals the number of connected components of , and any other eigenvalue satisfies the condition , for each .
2.1. Replacement product of graphs
The replacement product of two graphs is a simple and intuitive construction, which is well known in the literature, where it was often used in order to reduce the vertex degree without losing the connectivity property. It has been widely used in many areas including Combinatorics, Probability, Group theory, in the study of expander graphs and graph-based coding schemes [20, 21]. It is worth mentioning that Gromov studied the second eigenvalue of an iterated replacement product of a -dimensional cube with a lower dimensional cube .
Let us introduce some notation. Let be a finite connected -regular graph (loops and multi-edges are allowed). Suppose that we have a set of colors (labels), that we identify with the set of natural numbers . We assume that, for each vertex , the edges incident to are labelled by a color near , and that any two distinct edges issuing from have a different color near . A rotation map is defined by
if there exists an edge joining and in , which is colored by the color near and by the color near . We may have . Moreover, it follows from the definition that the composition is the identity map. Since an edge of joining the vertices and is colored by some color near and by some color near , we will say that the graph is bi-labelled.
Let be a connected -regular graph, and let be a connected -regular graph, satisfying the condition . The replacement product is the regular graph of degree with vertex set , that we can identify with the set , and whose edges are described by the following rotation map:
for all .
One can imagine that the vertex set of is partitioned into clouds, which are indexed by the vertices of , where by definition the -cloud, for , consists of vertices . Within this construction, the idea is to put a copy of around each vertex of , while keeping edges of both and . Every vertex of will be connected to its original neighbors within its cloud (by edges coming from ), but also to one vertex of a different cloud, according to the rotation map of . Note that the degree of depends only on the degree of the second factor graph .
Notice that the definition of depends on the bi-labelling of . In general, there may exist two different bi-labellings of , such that the associated replacement products are non isomorphic graphs [1, Example 2.3].
2.2. Zig-zag product of graphs.
The zig-zag product of two graphs was introduced in  as a construction which produces, starting from a large graph and a small graph , a new graph . This new graph inherits the size from the large graph , the degree from the small graph , and the expansion property from both graphs. The most important feature of the zig-zag product is that is a good expander if both and are; see Reingold, Vadhan, Wigderson [25, Theorem 3.2]. There it is explicitly described how iteration of the zig-zag construction, together with the standard squaring, provides an infinite family of constant-degree expander graphs, starting from a particular graph representing the building block of this construction.
Let be a connected -regular graph, and let be a connected -regular graph such that (as usual, graphs are allowed to have loops or multi-edges). Let (resp. ) be the rotation map of (resp. ). The zig-zag product is a regular graph of degree with vertex set , that we identify with the set , and whose edges are described by the rotation map
for all , if:
where , and .
Observe that labels in are elements of . As in the case of the replacement product, the vertex set of is partitioned into clouds, indexed by the vertices of . By definition the -cloud consists of vertices , for every . Two vertices and of are adjacent in if it is possible to go from to by a sequence of three steps of the following form:
a first step “zig”within the initial cloud, from the vertex to the vertex , described by ;
a second step jumping from the -cloud to the -cloud, from the vertex to the vertex , described by ;
a third step “zag”within the new cloud, from the vertex to the vertex , described by .
From the definition of the replacement and the zig-zag product it follows that the edges of arise from paths of length in of type:
a first step within one cloud (the zig-step);
a second step which is a jump to a new cloud;
a third step within the new cloud (the zag-step).
In other words, is a regular subgraph of the graph obtained by taking the third power of . This fact can be explicitly expressed in terms of normalized adjacency matrices. More precisely, let (resp. ) be the normalized adjacency matrix of the graph (resp. ), and suppose that . Then the normalized adjacency matrix of is (see ), with , where the symbol denotes the tensor product, or Kronecker product, and is the permutation matrix on associated with the map , i.e,
The matrix has exactly one entry in each row and each column and ’s elsewhere. Note that and are both symmetric matrices, due to the undirectedness of and . On the other hand, it is easy to check that the normalized adjacency matrix of is , and that the following decomposition holds:
where is the normalized adjacency matrix of a regular graph.
Note that the replacement product and the zig-zag product are defined for finite connected regular graphs and . It follows from the definition of replacement product that the graph is connected; on the other hand, the connectedness of and does not ensure the connectedness of the graph . One of the goals of the current work is to investigate this property for the zig-zag construction.
Recall that a complete bipartite graph is a graph whose vertex set can be partitioned into two subsets and such that, for every two vertices , , one has , but there is no edge joining two vertices belonging to the same subset . A complete bipartite graph is usually denoted by , if and .
The following basic result will be very useful for the rest of the paper. It shows that the graph consists of unions of special “elementary blocks”, each isomorphic to a complete bipartite graph.
Let be a -regular graph, and let be a -regular graph on vertices. Suppose that the vertices and are adjacent in , with , and . Let be the set of vertices adjacent to in ; similarly, let be the set of vertices adjacent to in . Then the edge connecting and in produces in the following subgraph isomorphic to :
The hypothesis ensures that the graphs and contain the subgraphs depicted below.
Now it suffices to apply the definition of the zig-zag product in order to get the assertion. ∎
In the case where is a -regular graph, for instance, if is a cycle graph, we call the graph the papillon graph. We will say that the vertices form the papillon graph depicted below.
Let be the -dimensional Hamming cube, so that is the set of binary words of length , and two words and are adjacent if and only if for all but one index . Observe that can be interpreted as the Cayley graph of the group with respect to the generating set , where denotes the triple with at the -th coordinate and elsewhere. Now let be the cycle graph of length , which can be interpreted as the Cayley graph of the cyclic group , with respect to the generating set (see Figure 1).
Having these interpretations in our mind, we label the edges of as follows: the edge connecting two vertices and is labelled by both near and , if the corresponding words and differ in the -th letter (this corresponds to moving by using the generator in the Cayley graph of ). Similarly, we label the edges of in such a way that , where the integers are taken modulo (this corresponds to moving by using the generators in the Cayley graph of ). In the replacement product, every vertex of is replaced by a cloud of vertices representing a copy of . Moreover, each vertex of any cloud is connected to exactly one vertex of a neighboring cloud according with the following rule: the vertex is connected to the vertex . The replacement product is depicted in Figure 2. It is worth mentioning that the replacement product is isomorphic to the Cayley graph of the semidirect product , with respect to the generating set (see ).
The zig-zag product is represented in Figure 3. It is known that it can be regarded as the Cayley graph of the group , with respect to the generating set (see ). Observe that one edge in this graph is obtained as a sequence of three steps in the graph in Figure 2: the first step within some initial cloud, the second step jumping to a new cloud and finally a third step within the new cloud. For instance, the three steps produce the edge connecting the vertices and in . The papillon subgraphs structure of is well-rendered in Figure 3.
In this section we discuss the connectedness problem of the zig-zag product of graphs. We will see (Example 5.3), that in general such graph product is not connected. The upcoming result relates the investigation of the connectedness properties of the graph to the study of a new graph that one constructs starting from , independently of .
Like before, let be a -regular connected graph and , with . For any , we put . Now we associate with a new graph , called the neighborhood graph of .
The neighborhood graph of is defined by:
In other words, two vertices of are adjacent if and have at least a common neighbor in . We do not put any label on the edges of .
Let be a -regular graph and let , with . If the neighborhood graph of is connected, then is connected as well.
It is enough to show that for all and , the vertices and are in the same connected component of . In fact, if are adjacent vertices in , there exist such that . This implies that the vertices and are connected in , for all and . Hence, if two vertices and are connected by a path in , there exists a path in connecting and , for suitable . By combining this property with the fact that and are in the same connected component of for every , we get the assertion.
So, let us show now that and belong to the same connected component. Since is connected, there exists a sequence such that , and , for . Let . Notice that and are connected in , since they have the common neighbor , where and . The same can be said for and , as they share a neighbor of the form where and . By using the same argument, we can say that and are in the same connected component of . This ensures that (and in particular and ) are in the same connected component of and this concludes the proof. ∎
From now on, the complete graph on vertices will be denoted by , and the cycle graph on vertices will be denoted by .
Let be a -regular graph. Then for every , the graph is connected. If is odd, then the graph is connected.
It suffices to observe that the neighborhood graph associated with is isomorphic to itself; similarly, it is straightforward to check that the neighborhood graph associated with is isomorphic to . ∎
Notice that the condition of the previous theorem is not a necessary condition. In the case of the Schreier graphs of the Basilica group discussed in Section 6, the zig-zag product is connected, for every , even if the neighborhood graph associated with the cycle graph consists of two connected components.
4. Isomorphism properties
In what follows we focus our attention on the case when the factor graph in the zig-zag product is a cycle graph. This assumption allows us to give precise results about the structure of the connected components of .
We have seen in Corollary 3.1 that if is an odd integer, then the zig-zag product is always connected, independently of the bi-labelling of the edges of . For this reason, our analysis will be restricted to the case , with an even .
4.1. Parity blocks
Let be a -regular bi-labelled graph, where is an even natural number; recall that an edge joining two vertices and in is colored by some color near and by some color near . As usual, we identify the set of colors with the set . Let (resp. ) be the subset of consisting of the even (resp. odd) numbers from to , so that . Given , and chosen one of the sets , , the parity block is the subgraph of defined as follows:
is the set of all vertices with the property that there exists a path in such that the following parity properties are satisfied:
, for ;
consists of the edges joining two consecutive vertices and in , and bi-labelled according with the bi-labelling of , described by the rotation map .
A vertex is said to be even or with parity (resp. odd or with parity ) in if the path is such that (resp. ). If a vertex is both even and odd, we will say that is odden or with parity . In other words, the vertex is odden if and only if and coincide.
Since is finite, decomposes into a finite number of ’s, in the sense that every edge in belongs to some graph for some and . We write where runs over an opportune finite index set. Notice that a vertex which is either even or odd in a parity block has degree in that parity block, whereas an odden vertex has degree in the parity block.
Let be a parity block of . If is a vertex in with parity in , then .
By definition has parity in if there exists a path in , with , such that and . In order to prove our statement, it is enough to show that . Notice that the inverse path , with , satisfies the parity conditions, with the property that with parity , as . ∎
This result ensures that, given a bi-labelled graph , a decomposition of into parity blocks is uniquely determined, and we are allowed to use the notation , without explicitly expressing the dependence of the parity blocks on the particular vertices. Given a bi-labelled graph , we will refer to its decomposition into parity blocks as its parity block decomposition. We will often write