Riordan graphs I: Structural properties^{1}
Abstract
In this paper, we use the theory of Riordan matrices to introduce
the notion of a Riordan graph. The Riordan graphs are a farreaching
generalization of the well known and well studied Pascal graphs and
Toeplitz graphs, and also some other families of graphs. The Riordan
graphs are proved to have a number of interesting (fractal)
properties, which can be useful in creating computer networks with
certain desirable features, or in obtaining useful information when
designing algorithms to compute values of graph invariants. The main
focus in this paper is the study of structural properties of
families of Riordan graphs obtained from infinite Riordan graphs,
which includes a fundamental decomposition theorem and certain
conditions on Riordan graphs to have an Eulerian trail/cycle or a
Hamiltonian cycle. We will study spectral properties of the Riordan
graphs in a follow up paper.
AMS classifications:
05C75, 05A15, 05C45
Key words: Riordan matrix, Riordan graph, Pascal graph, Toeplitz graph, fractal, graph decomposition
This paper is dedicated to the memory of Professor Jeff Remmel, who recently passed away.
1 Introduction
Pascal’s triangle is a classical combinatorial object, and its roots can be traced back to the 2nd century BC. In 1991, Shapiro, Getu, Woan and Woodson [15] have introduced the notion of a Riordan array, also known as a Riordan matrix, in order to define a class of infinite lower triangular matrices with properties analogous to those of the Pascal triangle (matrices) [11]. Since then, Riordan matrices became an active area of research. See [14] by Merlini and Sprugnoli, and references there in, for examples of results in this direction. Also, see [6] for a recent paper about Lie theory on the Riordan group, the set of invertible Riordan matrices. In particular, Riordan matrices found applications in the context of the computation of combinatorial sums [17].
The notion of Pascal’s triangle was also influential in graph theory and its applications to computer networks. Indeed, in 1983, Deo and Quinn [8] have introduced the Pascal graphs that are constructed using Pascal’s triangle modulo 2. These graphs attracted much attention in the literature (see [5] and references there in) and they are optimal graphs for computer networks with certain desirable properties, such as

the design is to be simple and recursive;

there must be a universal vertex, i.e. a vertex adjacent to all others;

there must exist several paths between each pair of vertices.
Another important object of interest to us is the well studied notion of a Toeplitz graph, that is based on the notion of a Toeplitz matrix (see [12] and references there in). A Toeplitz graph is a graph with and .
In this paper we introduce the notion of a Riordan graph. This notion not only provides a farreaching generalizaiton of the notions of a Pascal graph and a Toeplitz graph, but also extends the theory of Riordan matrices to the domain of graph theory, similarly to the introduction of the Pascal graphs based on Pascal’s triangles. The Riordan graphs are proved to have a number of interesting (fractal) properties, which can be useful in creating computer networks with certain desired features, or in obtaining useful information when designing algorithms to compute values of graph invariants. Indeed, the Pascal graphs are just one instance of a family of Riordan graphs having a universal vertex and a simple recursive structure. Thus, other members of that family could be used instead of the Pascal graphs in designing computer networks, and depending on the context, these could be a better choice than the Pascal graphs.
Our basic idea here is in building the infinite adjacency matrix based on an infinite Riordan matrix modulo 2, and considering the leading principal matrices giving (finite) Riordan graphs. See Section 2.2 for the precise definitions. We introduce various families of Riordan graphs based on the choice of the generating functions defining these graphs (via Riordan matrices). For example, the Riordan graphs of the Appell type are precisely the class of Toeplitz graphs including the Fibonacci graphs, while the Riordan graphs of the Bell type include the Pascal graphs, Catalan graphs and Motzkin graphs.
One of the basic questions one can ask in our context is whether or not a given labelled or unlabelled graph is a Riordan graph (defined by a pair of generating functions). It turns out that all unlabelled graphs on at most four vertices are Riordan graphs (see Figure 1), while nonRiordan unlabelled graphs always exist for larger graphs on any number of vertices. However, the main focus in this paper is the study of labelled Riordan graphs, and we give structural properties of certain families of graphs obtained from infinite Riordan graphs.
Throughout the paper, we normally label graphs on vertices by the elements of . However, we also meet graphs labelled by odd numbers, or even numbers, or consecutive subintervals in . For two isomorphic graphs, and , we write . For a graph , (resp., ) denotes the set of vertices (resp., edges) in . Also, for a subset of vertices in a graph , we let denote the graph induced by the vertices in . Moreover, for a formal power series , denotes the coefficient of in the sum. Finally, we let .
This paper is organized as follows. In Section 2 we review the notion of a Riordan matrix and use it to introduce the notion of an (infinite) Riordan graph in the labelled and unlabelled cases. However, the main focus in this paper is the labelled case, so unless we use the word “unlabelled” explicitely, our Riordan graphs are labelled. A number of basic results on Riordan graphs are established in Section 2.2, and this includes the number of Riordan graphs on vertices (see Proposition 2.3), and the necessary conditions on Riordan graphs (see Theorem 2.5). In Section 2.3 we define the product of two Riordan graphs and then give its combinatorial interpretation in terms of directed walks in certain graphs (see Theorem 2.11). We also discuss the ring sum of two graphs in Section 2.3 that can be used to define certain classes of Riordan graphs (see Section 2.4). Various families of Riordan graphs are introduced in Section 2.4. These include, but are not limited to Riordan graphs of the Appell type, Bell type, Lagrange type, checkerboard type, derivative type, and hitting time type.
In Section 3 we give structural results applicable to any Riordan graphs. In particular, in Section 3.1 we show that every Riordan graph is a fractal (see Theorem 3.6). Also, the reverse relabelling of proper Riordan graphs is defined and studied in Section 3.2. Further, in Section 3.3 we prove the Riordan Graph Decompostion theorem (see Theorem 3.12). In addition, in Section 3.4 we give certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle.
In Section 4.1, we consider iodecomposable and iedecomposable proper Riordan graphs, and provide a characterization result for these graphs (see Theorem 4.2). One of the main focuses in this paper is the study of Riordan graphs of the Bell type conducted in Section 4.2. In particular, we provide two characterization results for iodecomposable Riordan graphs of the Bell type (see Lemma 4.4 and Theorem 4.6) and use the results to show that the Pascal graphs and Catalan graphs are iodecomposable, while the Motzkin graphs are not iodecomposable. Also, in Section 4.2 we study the following properties of iodecomposable Riordan graphs of the Bell type: number of edges, number of universal vertices, clique number, chromatic number, diameter, and others. In Section 4.3 we provide two characterization results (Lemma 4.25 and Theorem 4.26) for iedecomposable Riordan graphs of the derivative type. Finally, in Section 5 we provide concluding remarks and state directions for further research.
We study spectral properties of the Riordan graphs in the follow up paper [4].
2 From Riordan matrices to Riordan graphs
After reviewing the notion of a Riordan matrix in Section 2.1, we will introduce the notion of a Riordan graph in Section 2.2. Then, in Section 2.4 we introduce various families of Riordan graphs.
2.1 Riordan matrices
Let be the ring of formal power series in the variable over an integral domain . If there exists a pair of generating functions , such that for ,
then the matrix is called a Riordan matrix (or, a Riordan array) over generated by and . Usually, we write . Since , every Riordan matrix is infinite and a lower triangular matrix. If a Riordan matrix is invertible, it is called proper. Note that is invertible if and only if , and .
If we multiply by a column vector with the generating function over an integral domain with characteristic zero, then the resulting column vector has the generating function . This property is known as the fundamental theorem of Riordan matrices (FTRM). This leads to the multiplication of Riordan matrices, which can be described in terms of generating functions as
(1) 
The set of all proper Riordan matrices under the above Riordan multiplication forms a group called the Riordan group. The identity of the group is , the usual identity matrix and where is the compositional inverse of , i.e. .
Throughout this paper, we write for .
For a Riordan matrix over , the matrix defined by
is called a binary Riordan matrix, and it is denoted by . The leading principal matrix of order in (resp., ) is denoted by (resp., ). If then the fundamental theorem gives
(2) 
It is known [13] that an infinite lower triangular matrix with is a proper Riordan matrix if and only if there is a unique sequence with such that, for ,
This sequence is called the sequence of the Riordan array. Also, if then
(3) 
where is the generating function of the sequence of . In particular, if is a binary Riordan matrix with then the sequence is called the binary sequence where .
2.2 Riordan graphs
The following definition gives the notion of a Riordan graph in both labelled and unlabelled cases. We note that throughout this paper the graphs are assumed to be labelled unless otherwise specified.
Definition 2.1
A simple labelled graph with vertices is a Riordan graph of order if the adjacency matrix of is an symmetric matrix given by
for some Riordan matrix over . We denote such by , or simply by when the matrix is understood from the context, or it is not important. A simple unlabelled graph is a Riordan graph if at least one of its labelled copies is a Riordan graph.
We note that the choice of the functions and in Definition 2.1 may not be unique. If is a Riordan graph and then for ,
(4) 
Thus the adjacency matrix satisfies that

its main diagonal entries are all 0, and

its lower triangular part below the main diagonal is the binary Riordan matrix .
For example, the Catalan graph where
is given by

Definition 2.2
A Riordan graph is proper if the binary Riordan matrix is proper.
If a Riordan graph is proper then the Riordan matrix is also proper because . The converse to this statement is not true. For instance, is a proper Riordan matrix but is not a proper Riordan graph.
For any Riordan graph , we can think of the sequence of induced subgraphs
each defined by the same pair of functions, showing the recursive nature of Riordan graphs. From applications point of view, this property implies that when a new node is added to a network, the entire network does not have to be reconfigured.
Proposition 2.3
The number of Riordan graphs of order is
Proof. Let be a labelled Riordan graph and be the smallest index such that .

If then is the null graph .

If then we can assume that and where .
Thus the number of possibilities to create is
where the 1 corresponds to the null graph.
Definition 2.4
Any Riordan matrix over naturally defines the infinite graph
which we call the infinite Riordan graph corresponding to the Riordan matrix .
We note that even if an unlabelled graph is Riordan, its random labelling is likely to result in a nonRiordan graph. The following theorem gives necessary conditions for a graph to be Riordan. These conditions are formulated in terms of the subdiagonal elements in the adjacency matrix of a Riordan graph.
Theorem 2.5 (Necessary conditions for Riordan graphs)
Let be a Riordan graph of order . Then one of the following holds:

for .

and for .

for , i.e. has the Hamiltonian path .
Proof. Let . From the definition of the Riordan matrix , we have
for where
and . Going through the four possibilities
of choosing and in
, we obtain the required result.
Figure 1 justifies that all unlabelled graphs on at most four vertices are Riordan (proper labelling and the corresponding Riordan matrices are provided in the figure). On the other hand, the following proposition shows that not all unlabelled graphs on vertices are Riordan for .
Proposition 2.6
The unlabelled graph obtained from a complete graph by adding an isolated vertex is not Riordan for .
Proof. Suppose that there exist and such that a labelled copy of is the Riordan graph . We consider two cases depending on whether the isolated vertex in is labelled by or not.
Let the isolated vertex be labelled by 1. Since there are no edges for , we have so that is the null graph . This is a contradiction.
Let be the label of the isolated vertex and . Since
by Theorem 2.5 this is also a contradiction. Hence the proof follows.
2.3 Operations on Riordan graphs
There are many graph operations studied in the literature. However, in general we cannot guarantee that a particular operation applied to Riordan graphs results in a Riordan graph. An example of an operation that can be used in our context is the ring sum of two graphs defined as follows.
Definition 2.7
Given two graphs and we define the ring sum . Thus, an edge is in if and only if it is an edge in , or and edge in , but not both.
The ring sum is well defined on Riordan graphs with a fixed and a fixed vertex set (e.g. ) due to the fact that , so
Next, we define a new graph operation, the product of two Riordan graphs, and then we give its combinatorial interpretation in terms of directed walks in certain graphs.
Definition 2.8
The product of two Riordan graphs and is the graph
(5) 
The set of all proper Riordan graphs forms a group under the binary operation given by (5), which follows from (1). The identity of the group is the path graph . The adjacency matrix , where is given by
Definition 2.9
Let and . We define the RGBgraph to be a digraph on the set of vertices with colored edges as follows:

For each edge in , , add the edge to and color it in Red.

For each edge in , , add the edge to and color it in Blue.

For each , , add the edge to and color it in Green.
The adjacency matrix of the RGBgraph is defined as the (0,1)matrix whose entry is 1 if and only if .
See Example 2.13 below for an instance of a graph .
Definition 2.10
An RGBwalk in is a directed walk of length in such that the first edge in it is Red, the second edge is Green, and the third edge is Blue.
Theorem 2.11
Let and . Then two vertices and in , for , are adjacent if and only if the number of RGBwalks in from to is odd.
Proof. Let , and be matrices. Thus, (resp., ; ) is the adjacency matrix of the directed subgraph of formed by the red (resp., green; blue) edges. All entries in are 0 except for . Thus if then
(6) 
It implies that if then counts the number of RGBwalks in the digraph from to . Now let be the adjacency matrix of . Since
we have
(7) 
Since for it follows from (6) and (7) that if then . It means that and for are adjacent, i.e. if and only if is odd. Hence the proof follows.
Remark 2.12
Let and . Since is the path graph, by Theorem 2.11 the number of RGBwalks from to is even (resp., odd) when (resp., ).
2.4 Families of Riordan graphs
Below, we introduce a number of classes of Riordan graphs and give examples of graphs in these classes. The names of the classes come from the widely used names of the Riordan matrices defining the respective Riordan graphs; such matrices are obtained by imposing various restrictions on the pairs of functions . Additionally, in Section 4 we introduce odecomposable, edecomposable, iodecomposable and iedecomposable Riordan graphs. Also, more classes of Riordan graphs can be introduced using the operations and defined in Section 2.3, and we discuss these at the end of this subsection.
Note that the most general definition of the null graphs
(also known as the empty graphs) in our terms is
for any where .
Also note that the empty graphs, the star graphs , and the complete ary trees for defined by are examples of nonproper Riordan graphs; other examples of nonproper Riordan graphs can be obtained from (v) in Theorem 3.14, and even more such examples are discussed at the end of this subsection. However, most of Riordan graphs considered in this paper are proper.
Riordan graphs of the Appell type. This class of graphs is defined by an Appell matrix , and thus it is precisely the class of Toeplitz graphs. Examples of graphs in this class are

the null graphs defined by ;

the path graphs defined by ;

the complete graphs defined by ;

the complete bipartite graphs defined by ; and

the Fibonacci graph defined by .
Riordan graphs of the Bell type. This class of graphs is defined by a Bell matrix . Examples of graphs in this class are

the null graphs defined by ;

the path graphs defined by ;

the Pascal graphs defined by ;

the Catalan graphs defined by ; and

the Motzkin graphs defined by .
Riordan graphs of the Lagrange type. This class of graphs is defined by a
Lagrange matrix
Riordan graphs of the checkerboard type. This class of graphs is defined by a checkerboard matrix such that is an even function and is an odd function. Examples of graphs in this class are

the path graphs defined by ; and

the complete bipartite graphs defined by .
Riordan graphs of the derivative type. This class of graphs is defined by a derivative matrix . Examples of graphs in this class are

the null graphs defined by ; and

the path graphs defined by .
Riordan graphs of the hitting time type. This class of graphs is defined by a hitting time matrix , , and it is trivially related to Riordan graphs of the derivative type. Indeed, the lower triangular part below the main diagonal of the adjacency matrix of any graph is
where the coefficients are taking modulo 2. Removing the first row and the first column of , corresponding to removing the vertex 1, gives the graph of the derivative type. Conversely, given a Riordan graph of the derivative type, one can relabel each vertex by , and add a new vertex, labelled by 1, that is connected to the vertices defined by the coefficients of the function to obtain .
Thus Riordan graphs of the Bell type (Section 4.2) and the derivative type (Section 4.3) are of the main interest in Section 4.
As is mentioned above, more classes of Riordan graphs can be introduced using the operations and defined in Section 2.3. Indeed, to illustrate this idea, note that the ring sum of a Riordan graph of the Bell type and the Riordan graph of the derivative type is well defined. Such a sum results in a new class of graphs defined by the Riordan matrices of the form . Indeed, since (mod 2), we have
Note that is not proper, as is the ring sum of any two proper Riordan graphs.
We end the subsection by noticing that the class of Riordan graphs of the Appell type (i.e. Toeplitz graphs) are closed under the operations and . It is not difficult to show that this class of graphs on vertices forms a commutative ring with the identity element and the zero element .
2.5 The complement of a Riordan graph
For any Riordan graph of the Appell type, the ring sum gives the complement of , i.e. the graph in which edges of become nonedges, and vice versa.
In general, it is not true that the complement of a Riordan (labelled or unlabelled) graph is Riordan. Indeed, the complement of the star graph , , is a labelled copy of the graph in Proposition 2.6, which is nonRiordan. Thus, Riordan graphs can become nonRiordan, and thus versa, under taking the complement.
Note that the complement of the Riordan graph in Figure 1 is the Riordan graph showing that the operation of the complement preserves the property of being Riordan for some graphs of nonAppell type.
3 Structural properties of Riordan graphs
We begin with basic properties of a Riordan graph , which can be directly determined in terms of binary column generating functions denoted of the binary Riordan array . Let
and let be the substitution of in the Taylor expansion in up to degree of modulo 2.
In this paper, denotes the degree of a vertex in a graph . If is understood from the context, we simply write .
Theorem 3.1 (Basic Properties)
Let be a Riordan graph. Then

For , if and only if .

.

.

For , .

.

If is proper then it has the Hamiltonian path . If additionally then has the Hamiltonian cycle .
Proof. The (i)–(iii) are straightforward. The (iv)
follows from the fact that the degree of a vertex is the
summation of all entries located in both the th column and
the th row of . The (v) follows from the
fact that the number of edges in is equal to the number
of 1s in . The (vi) follows from the fact that
if is proper then all entries of the subdiagonal in its
adjacency matrix are 1s, i.e. is
adjacent to for .
In what follows, the matching number is the size of a maximal matching in a graph .
Theorem 3.2
Let be a proper Riordan graph. Then .
Proof.
By (vi) in Theorem 3.1, has the Hamiltonian path . Therefore, for even and odd , respectively, maximal matchings are and . This completes the proof.
3.1 Fractal properties of Riordan graphs
A fractal is an object exhibiting similar patterns at increasingly small scales. Thus, fractals use the idea of a detailed pattern that repeats itself.
In this section, we show that every Riordan graph with has fractal properties by using the notion of the sequence of a Riordan matrix.
Definition 3.3
Let be a graph. A pair of vertices in is a cognate pair with a pair of vertices in if

and

is adjacent to is adjacent to .
The set of all cognate pairs of is denoted by cog.
Definition 3.4
The sequence of the binary Riordan matrix defining a Riordan graph with is called the binary sequence of the graph.
The following theorem gives a relationship between cognate pairs and the sequence of a Riordan graph.
Theorem 3.5
For , let be the binary sequence for a Riordan graph where and . Then
where for integers and .
Proof. Let be the adjacency matrix of . Without loss of generality, we may assume that . By Lemma 3.11, (3) and (4), we obtain
(8) 
where , . Since for it follows that if and only if
(9) 
Thus we obtain the desired result.
In particular, if is adjacent to then the pairs cognate with are those connected by edges in the following figures:

for 

