Permutational powers of a graph

This paper introduces a new graph construction, the permutational power of a graph, whose adjacency matrix is obtained by the composition of a permutation matrix with the adjacency matrix of the graph. It is shown that this construction recovers the classical zig-zag product of graphs when the permutation is an involution, and it is in fact more general. We start by discussing necessary and sufficient conditions on the permutation and on the adjacency matrix of a graph to guarantee their composition to represent an adjacency matrix of a graph, then we focus our attention on the cases in which the permutational power does not reduce to a zig-zag product. We show that the cases of interest are those in which the adjacency matrix is singular. This leads us to frame our problem in the context of equitable partitions, obtained by identifying vertices having the same neighborhood. The families of cyclic and complete bipartite graphs are treated in details.


Introduction
Graphs are among the most popular and useful tools used in Mathematics to model aspects of real life. Their relatively simple and natural definition makes these objects very versatile and adaptable in many areas of mathematical research. Usually graphs are studied via their adjacency matrix, and it is an interesting task to investigate the relationship between the geometrical properties of a graph and the algebraic properties of the corresponding adjacency matrix. Moreover, there exists a correspondence between operations that can be performed on graphs and operations on matrices. See, for example, [5,7,11,12,16]. In the last years, a very intriguing product of graphs that has been introduced is the so called zig-zag product: iterated applications of this product allow to construct infinite families of expanders.
The zig-zag product of two graphs was introduced in [15] as a construction allowing to produce, starting from a large graph G and a small graph H, a new graph G z H which inherits the size from the large one, the degree from the small one, and the expansion property from both the graphs. In [15], it is explicitly described how iteration of this construction, together with the standard squaring, provides an infinite family of constantdegree expander graphs, starting from a particular graph representing the building block of this construction (see [13] for further details on expanders). Topological properties of the zig-zag product have been studied in the paper [6]. It is worth mentioning that the zig-zag product has also interesting connections with Geometric group theory, as it is true that the zig-zag product of the Cayley graphs of two groups returns the Cayley graph of the semi-direct product of the groups [1]. See also the book [3] for connections of the zig-zag product with other graph compositions and random walks.
In the present paper, we show that one can in any case replace the role of G by an appropriate permutation matrix P of order 2. In other words, if G has k vertices, one can realize the adjacency matrix of the zig-zag product G z H as the productÃ H PÃ H , where A H is the adjacency matrix of the graph H (so thatÃ H = I k ⊗ A H can be regarded as the adjacency matrix of the graph obtained by taking k ≥ 1 disjoint copies of H), and P is a symmetric permutation matrix. Keeping this in mind, one can ask if it is possible to get the adjacency matrix of some graph (i.e., a symmetricÃ H PÃ H ) by considering also permutation matrices P that are not symmetric. The permutational power graph corresponds exactly to this case. Now it is clear that, whenever we havẽ A H PÃ H symmetric (and so it can interpreted as the adjacency matrix of an undirected graph) we can ask if the same graph can be obtained via the "classical" zig-zag product (by using a symmetric P ). This problem is strictly related to the singularity of the matrix A H . In the case in which A H is invertible, there is no chance to get a symmetricÃ H PÃ H with a nonsymmetric P . For this reason, we focus our attention on the cases where A H is singular.
In fact, the reason why the graph H is singular says a lot about the nature of its permutational power. The most general case is when we only know that the adjacency matrix of H is singular. We consider this case in Section 3.1, proving that a permutation p induces a permutational power of H if its projection on the range of the adjacency matrix of H is symmetric. Moreover, a permutational power of H with respect to p can be obtained by a classical zig-zag product if there exists an involution q sharing with p the same projection on the range of the adjacency matrix of H. One of the reasons why the adjacency matrix of H may be singular is that there are vertices sharing the same neighborhood: this is one of the cases analyzed in Section 4 via the notion of neighborhood equitable partition. Actually, in the framework of random matrices and random graphs, it is conjectured that this is the generic reason why H could be singular (for the symmetric case see [4]). The graph obtained collapsing the equivalent vertices together (in a precise way), can be singular or not. In the latter case every permutational power of H can be obtained via the classical zig-zag product (Theorem 4.13): the easiest examples are the complete bipartite graphs. On the other extreme, the adjacency matrix of H can be singular even if there are no vertices sharing the same neighborhood: the easiest examples are cyclic graphs C n with n divisible by 4.
In this paper we show, among other results, that: • the symmetry ofÃ H PÃ H only depends on the symmetry of the projection of P on the range of the matrixÃ H (see Theorem 3.7 and Corollary 3.8); • there are permutational powers that do not appear as classical zig-zag products (see Corollary 3.12); • we can restrict our attention, in a specific sense, to graphs whose adjacency matrix is singular with pairwise distinct rows (see Corollary 4.7, Corollary 4.8 and Theorem 4.13).
Moreover, the cases in which H is a cycle or a complete bipartite graph are studied in details (see Sections 3.2 and 3.3, and Section 4.1).

Preliminaries and motivations
In this paper we will denote by G = (V G , E G ) a finite undirected graph with vertex set V G and edge set E G . If {u, v} ∈ E G , we say that the vertices u and v are adjacent in G, and we write u ∼ v. Observe that loops and multi-edges are allowed. A path of length t in G is a sequence u 0 , u 1 , . . . , u t of vertices such that u i ∼ u i+1 . The graph is connected if, for every u, v ∈ V G , there exists a path u 0 , u 1 , . . . , u t in G such that u 0 = u and u t = v.
Suppose |V G | = n. We denote by A G = (a u,v ) u,v∈V G the adjacency matrix of G, i.e., the square matrix of size n whose entry a u,v is equal to the number of edges joining u and v. The degree of a vertex u ∈ V G is defined as deg(u) = v∈V G a u,v . In particular, we say that G is regular of degree d, or d-regular, if deg(u) = d, for each u ∈ V G . For such a graph G, the normalized adjacency matrix is defined as We recall now the definition of the zig-zag product of two graphs. This is a noncommutative graph product producing a graph whose degree depends only on the degree of the second component graph, and providing large and sparse graphs. We need some notation.
Let G = (V G , E G ) be a d-regular graph over n vertices. Suppose that we have a set of d colors (labels), that we identify with the set [d] = {1, 2, . . . , d}. We can assume that, for each vertex v ∈ V G , the edges incident to v are labelled by a color h ∈ [d] near v, and that any two distinct edges issuing from v have a different color near v. This allows to define the rotation map Rot G : if there exists an edge joining v and w in G, which is colored by the color h near v and by the color k near w. Note that it may be h = k. Moreover, the composition Rot G • Rot G is the identity map.
, and whose edges are described by the rotation map Observe that labels in G z H are elements from [d H ] 2 . The vertex set of G z H is partitioned into n clouds, indexed by the vertices of G, where by definition the v-cloud, with v ∈ V G , is constituted by the vertices (v, 1), (v, 2), . . . , (v, d G ). Two vertices (v, k) and (w, l) of G z H are adjacent in G z H if it is possible to go from (v, k) to (w, l) by a sequence of three steps of the following form: (1) a first step "zig"within the initial cloud, from the vertex (v, k) to the vertex (v, k ′ ), described by Rot H (k, i) = (k ′ , i ′ ); (2) a second step jumping from the v-cloud to the w-cloud, from the vertex (v, k ′ ) to the vertex (w, l ′ ), described by Rot G (v, k ′ ) = (w, l ′ ); (3) a third step "zag"within the new cloud, from the vertex (w, l ′ ) to the vertex (w, l), described by Rot H (l ′ , j) = (l, j ′ ).
Example 2.2. Consider the graphs G and H in Fig. 1. It follows from Definition 2.1 that the graph G z H, which is depicted in Fig. 2, is the graph with vertex set {1, 2, 3, 4} ×{a, b, c}, whose edges are labelled by ordered pairs (i, j) from {A, B} 2 . If we ask, for instance, which are the vertices adjacent to (1, a) in G z H, we have to take into account that: In this way, we conclude that the vertex (1, a) in G z H is adjacent to the vertices (3, a), (3, b), (4, b), (4, c).
Let us analyze the adjacency matrix of G z H. Let A G (resp. A H ) be the adjacency matrix of the graph G (resp. H). Let I n denote the identity matrix of size n, for each n ≥ 1. It follows from the definition of zig-zag product that the adjacency matrix of PSfrag replacements (1, a) [15]), where P G is the permutation matrix of size |V G ||V H | associated with the map Rot G , i.e., If G is the graph of Example 2.2, we have Observe that P 2 G = I 12 , as the rotation map is an involution. Equivalently, P G is a symmetric matrix.
For each positive integer k, putÃ H = I k ⊗ A H (observe that, for k = 1, one has A H = A H ). Our paper moves from the following remark: if we are given a graph H, and we consider k copies of such graph, and we are also given a symmetric permutation matrix P of size k|V H | (i.e., a matrix corresponding to a permutation of order 2 of k|V H | elements), then the symmetric matrix M =Ã H PÃ H can be regarded as the adjacency matrix of a graph composition of type "zig-zag", where the jump steps are codified by the matrix permutation P . In other words, the first factor graph is not essential to perform the zig-zag construction, one just needs the permutation matrix P describing the rotation map.

Permutational powers of a graph
The concluding remark of Section 2 can be formalized as follows. For every n ≥ 1, let Sym(n) denote the symmetric group on n elements. Take a regular graph H on m vertices, and fix a positive integer k ≥ 1. Let P be a symmetric matrix permutation on km elements, that is, the permutation p ∈ Sym(km) associated with P is the composition of disjoint transpositions (with possibly some fixed elements). Let us identify V H with the set {0, 1, . . . , m − 1}, and similarly identify the km vertices obtained by taking k copies of H with the set {0, 1, . . . , km − 1}. Observe that each number x ∈ {0, 1, . . . , km − 1} admits a unique representation as with i = 0, 1, . . . , k − 1 and j = 0, 1, . . . , m − 1.
With this interpretation, we can think that the vertex x is the j-th vertex belonging to the i-th copy of the graph H. Let us construct now a labelled m-regular graph G on k vertices as follows. Vertices will be named 0, 1, . . . , k − 1, whereas edges will have labels (colors) 0, 1, . . . , m − 1 around each vertex. More precisely, if the transposition τ = (s t) appears in p, with s = i s m + j s and t = i t m + j t , then we connect the vertices i s and i t in G by an edge labelled j s near the vertex i s and by j t near the vertex i t . If u = i u m + j u is an element fixed by p, there will be a loop at the vertex i u with two labels equal to j u . In this situation, the graph with adjacency matrixÃ H PÃ H coincides with the graph G z H, where G has been constructed as described above.

Remark 3.2.
Even if the zig-zag product has been defined in [15] for regular graphs, in order to construct increasing sequences of regular expander graphs, our construction shows that one can also start from a nonregular graph H. In fact, if one takes k copies of H and a permutation matrix on k|V H | elements such that the product M =Ã H PÃ H is symmetric, then M can be regarded as the adjacency matrix of a graph obtained from H by a composition of type "zig-zag". Example 3.3. In Fig. 5, two copies of a nonregular graph H on 6 vertices, denoted H 0 and H 1 , are represented. Take the permutation p = (0 4 6 3 7 5 1 8)(2 9)(10 11) on 12 elements. Construct the graph G associated with p (see Fig. 4). The graph resulting from the construction described above appears in the bottom of Fig. 5. Notice that it is not a regular graph. If, for instance, we ask which are the vertices in G z H which are adjacent  to the vertex 2, we have to think that 2 is a vertex belonging to the copy indexed by 0, and its neighbors are the vertices 0, 1, 3. Such vertices are mapped by p, respectively, to the vertices 4, 8, 7. Now, the only neighbor of 4 in the copy H 0 is 3; the neighbor of 7 in H 1 is 8; finally, the neighbors of 8 in H 1 are the three vertices 6, 7, 9. We then conclude that the vertices adjacent to 2 are exactly the vertices 3, 8, 6, 7, 9.
Notice that in Example 3.1 the graph G constructed starting from the permutation is undirected, whereas it is directed in Example 3.3, due to the fact that the order of the permutation is not 2. In Fig. 4 the labels in the arc directed from the copy 0 to the copy 1 must be interpreted as follows: 4, 0 corresponds to the fact that p(4) = 6 · 1 + 0 = 6; 3, 1 to the fact that p(3) = 6 · 1 + 1 = 7 and 1, 2 to the fact that p(1) = 6 · 1 + 2 = 8. Similarly, the loop at the 0 copy starting with 5 and ending with 1 corresponds to the fact p(5) = 6 · 0 + 1 = 1. The undirected edge and the undirected loops correspond to the transpositions (2 9) and (10 11), respectively.
One can also ask what happens when the matrixÃ H PÃ H is not symmetric: in this situation, the resulting matrix can be regarded as the signed adjacency matrix of a directed graph, what leads to the possibility of defining a zig-zag product of directed graphs, containing the classical product as a particular case. This general situation will not be investigated in the present paper.
On the other hand, it may happen that, even if the permutation p is not of order 2, anyway one has directed edges from each vertex in the neighborhood of v to each vertex in the neighborhood of w and viceversa, producing an undirected graph. This is the case we are interested in. The main questions that we address in our paper are the following.
(1) Given a graph H, with adjacency matrix A H , and taken a positive integer k, is it possible to find a nonsymmetric permutation matrix P on k|V H | elements such that the matrixÃ H PÃ H is symmetric? (2) If this is the case, under which conditions there exists a symmetric permutation matrix Q such thatÃ H PÃ H =Ã H QÃ H ? In other words, when such a permutational power of H can be obtained by the classical zig-zag product?
As an example, observe that the resulting graph in Fig. 3 can also be obtained by choosing the permutation p ′ = (0 11 3 6 10 7 8 4 1 9 2 5) on 12 elements. However, we will see that there exist permutational powers of graphs which cannot be obtained by the classical zig-zag construction of Definition 2.1 (see Corollary 3.12).
3.1. An algebraic interpretation. Let us start an algebraic investigation of the symmetry condition and so the graph with adjacency matrixÃ H PÃ H is a zig-zag construction.
By virtue of Lemma 3.5, the interesting cases that it is worth investigating are given by graphs H whose adjacency matrix is singular. Examples of such graphs are [2]: • the cyclic graph C n on n = 4h vertices; • the path graph P n on n = 2h + 1 vertices; • all bipartite graphs with an odd number of vertices; • graphs with two or more vertices sharing the same neighborhood (e.g., the complete bipartite graph K m,n ).
Then we have: If we repeat the same computation for the entry M y,x , we deduce that Eq. (1) is satisfied if and only if, for each i x , i y ∈ {0, 1, . . . , k − 1} and j x , j y ∈ {0, 1, . . . , m − 1}, one has: In other words, the number of the neighbors j l of the vertex j x in the copy i x of H such that p maps i x m + j l to i y m + j l ′ , where j l ′ is a neighbor of the vertex j y in the copy i y , must be equal to the number of the neighbors j l ′ of the vertex j y in the copy i y of H such that p maps i y m + j l ′ to i x m + j l , where j l is a neighbor of the vertex j x in the copy i x .
By a similar argument, one can prove the following proposition.
Proposition 3.6. Let H be a graph. Let k be a positive integer, and suppose that P is a permutation matrix of size k|V H | such that the matrixÃ H PÃ H is symmetric. Then also the matrixÃ H P −1Ã H is symmetric.
The same argument does not apply to the whole cyclic group generated by the permutation P , as there exist explicit examples showing that, if the matrixÃ H PÃ H is symmetric, then the matrixÃ H P hÃ H needs not to be symmetric for any integer h.
The matrix A H is symmetric, so that it admits all real eigenvalues, and an orthonormal basis of eigenvectors. Moreover, since we are dealing with a singular matrix, we can assume that 0 is an eigenvalue for A H . Put λ 0 = 0 and let λ 1 , . . . , λ s be the nonzero eigenvalues of A H . Let m i be the multiplicity of λ i , for each i = 0, 1, . . . , s, and let us denote by E i the corresponding eigenspace, so that E 0 = ker A H . Now, the spectrum of the matrixÃ H coincides with the spectrum of A H , but the eigenvalue λ i has multiplicity km i , for each i = 0, 1, . . . , s. The corresponding eigenspace is E i = R k ⊗ E i . Notice that, due to the symmetry of A H , we have where the eigenspaces E i are pairwise orthogonal. Sincẽ Similarly, given a permutation matrix P such thatÃ H PÃ H is symmetric, and whose order is not 2, a permutation matrix Q satisfiesÃ H PÃ H =Ã H QÃ H if and only if the Taking an orthonormal basis from each eigenspace E i , we construct an orthogonal matrix U such that U TÃ H U is diagonal. Put where U 1 is the submatrix whose columns form a basis of the space s i=1 E i , which is orthogonal to E 0 , and U 0 is the submatrix whose columns form a basis of E 0 . Theorem 3.7. For any two permutation matrices P and Q we havẽ This is equivalent to ask that, for every two columns u, v in U 1 , one has that (P − Q)u is orthogonal to v. Therefore, it must be U T 1 (P − Q)U 1 = O, that is the claim. Proof. We can apply Theorem 3.7 to the case Q = P T .
From an algebraic point of view, Theorem 3.7 and Corollary 3.8 give the complete answers to our two main questions, and we are going to apply them, in the next subsections, to the case of cyclic graphs. In Section 4, we will be interested in finding more geometric conditions, that in particular cases (e.g., complete bipartite graphs) characterize the permutations p inducing a permutational k-th power.
3.2. Cyclic graphs. In this section we are going to completely characterize the permutational 1-st powers of the cyclic graph C n . The adjacency matrix of C n is the circulant matrix Let us denote with R n the set of complex n-th roots of 1. It is a classical fact [2] that, for every λ ∈ R n , the vector v λ = (λ, λ 2 , . . . , λ n−1 , 1) is eigenvector for A Cn of eigenvalue λ +λ.
Consider a permutation p ∈ Sym(n) and the associated permutation matrix P . We are going to investigate under which conditions the matrix A Cn P A Cn is symmetric. As we already noticed, if n is not divisible by 4, the matrix A Cn is invertible and so by Lemma 3.5 the matrix P must be symmetric. From now on, assume that n = 4k. In the set [n] we define an involution i → i * , where i * is the element such that |i − i * | = 2k. Since n is even, we have −λ ∈ R n and λ i * = −λ i . Lemma 3.9. Let ζ be an n-th primitive root of 1, and let i 1 , i 2 , i 3 , i 4 ∈ [n]. Then: Proof. Set λ 1 = ζ i 1 , λ 2 = ζ i 2 , λ 3 = ζ i * 3 , λ 4 = ζ i * 4 , then we have λ 1 + λ 2 + λ 3 + λ 4 = 0, with λ 1 , λ 2 , λ 3 , λ 4 roots of unity. As a consequence of Theorem 6 in [14], the only possibility is that ( In the spirit of the introductory remarks to Theorem 3.7, the matrix A Cn P A Cn is symmetric if and only if (P − P T )v λ ∈ ker A Cn , for each λ ∈ R n \ {±i}, since the vectors {v λ : λ ∈ R n , λ = ±i} are a basis for the range of A Cn . Now we have: (2) . . .
We are now in position to prove the following theorem. In other words, there is no permutational 1-st power of C n with respect to a nonsymmetric permutation.
Observe that the graph C 4 belongs to the class of complete bipartite graphs, that we are going to analyze in detail in Section 4.1. The remaining case is the cycle graph C 8 , which is studied in the next section.
3.3. The cycle C 8 . In order to obtain necessary conditions for the symmetry of the matrix A C 8 P A C 8 , we study the behavior of the permutation p 2 . From the first line of Eq.
(2) and the first line of Eq. (3) we have that if i ≡ j mod 2 then Then if A C 8 P A C 8 is symmetric, we only have the following 4 possibilities: The case a) concerns permutations of order 2 and then the classical zig-zag product. In the case b), the even numbers are in 1-cycles or 2-cycles and there is a 4-cycle containing the odd numbers. We know that p(1) = p −1 (1). Considering i = 1 and j = 3 in Eq. By an explicit computation (our conditions, a priori, are only necessary) one can check that A C 8 P A C 8 is symmetric when p = σ i (and therefore also when p is a product of σ i with a permutation of order 2 of the even numbers).
An analogous argument in the case c) gives So we have that p is a permutation of case c) if and only if p is the product of a τ i with a permutation of order 2 of the odd numbers.
In the case d) we can apply Eq. (2) to all pairs (i, i * ), obtaining (6) p(i) = (p −1 (i * )) * , ∀i ∈ [n]; that is, p is conjugated to its inverse by the permutation induced by the involution * . Moreover, we have that Eq. (4) and Eq. (5) hold. In particular p 4 should be the identity or an involution and therefore p is an 8-cycle or the product of two 4-cycles. Moreover, if p were a 8-cycle, we would have that p 4 (i) = i * for any i ∈ [n], that is p = (a b c d a * b * c * d * ) for some a, b, c, d ∈ [n]. By applying Eq. (6) to p we have (a b c d a * b * c * d * ) = (d c b a d * c * b * a * ) that it is impossible. Thus p is a product of two disjoint 4-cycles. If the permutation sends even (resp. odd) numbers in even (resp. odd) numbers, we are just in the case p = τ i σ j . If this is not the case, the cycles alternate odd with even numbers; by Eq. (4) and Eq. (5)  By a direct computation, one can check that all these permutations make A C 8 P A C 8 symmetric.
By summarizing, the permutations p ∈ Sym(8) such that p 2 = Id and A C 8 P A C 8 is symmetric are the following 112 permutations: where i, j = 1, . . . , 4, k = 1, . . . , 16, and q (resp. s) is a permutation of order at most 2 fixing each odd (resp. even) number.

Equitable partitions
In this section we will use the notion of equitable partition. The main idea is that, whenever we declare equivalent vertices of the graph sharing the same set of neighbors,  Figure 6. The graph C 8 and its permutation 1-st power with respect to p 1 .
we get a partition of the vertex set which is equitable. This observation allows us to deeply explore the structure of the permutations giving rise to permutational powers and, in particular, to detect those permutational powers that actually can be obtained by a classical zig-zag product. Equitable partitions have a number of significant applications in Graph theory: for example, the vertex set partition of a graph under the action of a group of automorphisms is always equitable. This fact has been used in the context of graph isomorphism algorithms (we refer to the book [9] for more details).
Let G = (V G , E G ) be a graph, with |V G | = n. With a given partition π = {C 1 , . . . , C m } of V G we can associate an n × m matrix M π , called the characteristic matrix of π, defined as follows. For each v ∈ V G , and i ∈ {1, . . . , m}, one has It is easy to check that: Moreover we can use the characteristic matrix of π to define the m × m matrix A G /π = M T π A G M π . This matrix represents the restriction of the matrix A G to the parts of π.
It is known that the equitability condition of Definition 4.1 is equivalent to each of the following (the reader can refer to [8,9,10]): Moreover, if π is equitable, one has so that the entry (A G /π) i,j equals the total number of edges connecting a vertex of C i to a vertex of C j , multiplied by . In other words, the matrix A G /π can be regarded, up to a suitable normalization, as the adjacency matrix of the quotient graph G/π obtained from G by taking the quotient of V G modulo the equivalence relation defined by π.
In what follows we will mostly focus on a very specific equitable partitionπ, the one induced by the relation "to have the same neighborhood". This partition is natural in the context of graphs and fits into our setting of permutational powers of G. It concretely corresponds to the existence of two or more rows in the matrix A G which are equal. This condition assures that A G is not invertible and this is exactly the case we are interested in, by virtue of Lemma 3.5.
By definition,π is an equitable partition; we will call it the neighborhood partition of V G . The following proposition shows thatπ has an even stronger property.
Proof. For all u, v ∈ V G , we have: |Cu,v| if u and v belong to the same class C u,v ; 0 otherwise.
This implies: where we have used that a w,v = a u,v , because w and u are in the same part ofπ. Sinceπ is equitable, we conclude that A G MπM T π = MπM T π A G = A G .
In particular G/π is the graph where the vertices of G with the same neighborhood are identified.
A direct computation gives which is, up to normalization, the adjacency matrix of the quotient graph H/π in Fig. 7. PSfrag replacements We focus now our attention on the case of a graph H for which we want to investigate permutational k-th powers. Observe that the neighborhood partitionπ can be considered also for the graph obtained by taking k disjoint copies of H, and it is induced in a very natural way by the neighborhood partition of the vertex set V H . As usual, we put A H = I k ⊗ A H , and P is a matrix permutation acting on k|V H | elements, with k ≥ 1. Since a r,y l = a s,y l by hypothesis, we get the claim.
In Example 3.3 (Fig. 5), observe that the vertices 0 and 1, and the vertices 4 and 5, have the same neighborhood in the graph H: this implies that the pairs of vertices 0 and 1; 4 and 5; 6 and 7; 10 and 11, have the same neighborhood in the permutational power of H.
On the other hand, the condition of Proposition 4.5 is only sufficient, as it may occur that in the final graph two vertices share the same neighborhood, but the same property does not hold in the original graph. In Fig. 8, we have represented the cyclic graph C 8 , and its permutational 1-st power produced by the permutation p = (0 3 2 1)(4 5 6 7). One can directly check that the vertices 0 and 4 have the same neighborhood in the final graph, as well as the vertices 3 and 7; however, the equitable partition given by the neighborhood in C 8 is trivial, because there are no vertices sharing the same neighborhood in C 8 . Given a matrix permutation P and the partitionπ, we define P/π = M T π P Mπ in order to describe the action of the permutation on the classes induced byπ. We have the following reduction result.
Proof. It is a particular case of Theorem 4.6 when P 1 = P and P 2 = P T . Corollary 4.8. If P 1 /π = P 2 /π thenÃ H P 1ÃH =Ã H P 2ÃH . In particular, if P/π is symmetric thenÃ H PÃ H is symmetric.
The following proposition answers our Question (2) in Section 2, in the context of the neighborhood partitionπ. Proof. Letπ = {C 1 , . . . , C m } be the neighborhood partition on the vertex set of k disjoint copies of H, naturally induced by the neighborhood partition of V H . Notice that the entry (P/π) i,j counts, up to the factor 1/ |C i ||C j |, the number of elements in the class C i moved by the permutation p to the class C j . We define for any i, j ∈ {1, . . . , m}. Therefore (P/π) i,j = (P/π) j,i if and only if |V i,j can be empty. Since P/π is supposed to be symmetric, we have |V j,i |, so that we can define a bijection σ between these two sets. Since we can do this for any i and j we can extend the bijection to any of the parts C i . If σ(x) = y then we put q(x) = y and q(y) = x. On the sets V (p) i,i we can choose σ to be the identity map. The q we get is a permutation of order 2 on the set of vertices. It is clear that, in this way |V j,i | and this means that P/π = Q/π. The statement follows by applying Theorem 4.6.
Remark 4.10. The converse of Corollary 4.8 is false. In fact, we have seen in Example 3.11 that, in the case of the cycle C 8 , the partitionπ is trivial, so that P/π = P for any matrix permutation P , but we showed that there exist nonsymmetric permutation matrices P such that A C 8 P A C 8 is symmetric. Moreover, the converse of Proposition 4.9 is false, since we showed in Example 3.11 that there exist nonsymmetric permutation matrices P and symmetric permutation matrices Q such that A C 8 P A C 8 = A C 8 QA C 8 .
By arguing as in the proof of Proposition 4.9, we are able to give an estimate of the number of permutations of order 2 giving rise to the same graph in the case in which the matrix P/π is symmetric. Put p i,j = |V  Proof. Since the matrix P/π is symmetric, we have p i,j = |V which is a symmetric matrix. By virtue of Proposition 4.9, there exists a matrix permutation Q of order 2 such that ( One can directly check that a matrix Q of order 2 satisfying this property is the permutation matrix associated with the permutation q = ( 0 4)(1 8)(2 9)(3 7)(5 6)(10 11). Proof. IfÃ H PÃ H is symmetric, then by Corollary 4.7 the matrix (Ã H /π)(P/π)(Ã H /π) is symmetric; now, sinceÃ H /π is invertible, we deduce that P/π is symmetric. We can now apply Proposition 4.9 to get the claim.
Our results are useful when we analyze graphs with an adjacency matrix whose singularity depends on the repetition of some rows. The extreme example is treated in the following section.
4.1. The complete bipartite graph. Let K m,n denote the complete bipartite graph on m + n vertices, whose vertex set is partitioned into two sets V 1 and V 2 , with |V 1 | = m and |V 2 | = n, such that every vertex of V i is connected by an edge to every vertex of V j , with i = j, and no edge connects vertices belonging to the same part. Let J m,n denote the m × n matrix whose entries are all equal to 1. Then, up to a reordering of the vertices, the adjacency matrix of K m,n is M T π A Km,n Mπ = 0 √ mn √ mn 0 .
The latter matrix can be regarded, up to normalization, as the adjacency matrix of the graph K m,n /π, which reduces to the complete graph on 2 vertices. Now let k ≥ 1 be a positive integer and consider the disjoint union of k copies of K m,n , so that the adjacency matrix of this new graph is given byÃ Km,n = I k ⊗ A Km,n , and the associated characteristic matrix isMπ = I k ⊗ Mπ. In particular: Now let p be a permutation on k(m + n) elements, and let P be the corresponding permutation matrix. As the matrixÃ Km,n /π is nonsingular, by virtue of Corollary 4.7, the matrixÃ Km,n PÃ Km,n is symmetric if and only if the matrix P/π =M T π PMπ is symmetric. Therefore, by applying Proposition 4.9, we get the following theorem.
Theorem 4.14. A permutational k-th power of the graph K m,n can always be obtained by a classical zig-zag product. That is, if a permutation matrix P is such thatÃ Km,n PÃ Km,n is symmetric, then there exists a symmetric permutation matrix Q such thatÃ Km,n PÃ Km,n = A Km,n QÃ Km,n .
In order to investigate the symmetry of P/π, observe that P/π is a square matrix of size 2k, whose rows and columns can be indexed by the pairs (x, y), where the copy index x varies in {1, . . . , k} and the part index y varies in {1, 2}. Therefore, one has symmetry if and only if the number of elements in the part y of the copy x which are sent to elements in the part y ′ of the copy x ′ equals the number of elements in the part y ′ of the copy x ′ which are sent to elements in the part y of the copy x. Here below an explicit example in the case k = 2, m = 3, n = 5. with associated permutation matrix P . It is easy to check that: and therefore the matrix (I 2 ⊗ A K 3,5 )P (I 2 ⊗ A K 3,5 ) is symmetric. A permutation Q of order 2 satisfying the property (I 2 ⊗ A K 3,5 )P (I 2 ⊗ A K 3,5 ) = (I 2 ⊗ A K 3,5 )Q(I 2 ⊗ A K 3,5 ) is Q = (1 5)(2 11)(3 16)(6 9)(7 12)(8 14)(10 13), constructed as in the proof of Proposition 4.9.