#### Abstract

We give a recursion formula to generate all the equivalence classes of connected graphs with coefficients given by the inverses of the orders of their groups of automorphisms. We use an algebraic graph representation to apply the result to the enumeration of connected graphs, all of whose biconnected components have the same number of vertices and edges. The proof uses Abel’s binomial theorem and generalizes Dziobek’s induction proof of Cayley’s formula.

#### 1. Introduction

As pointed out in , generating graphs may be useful for numerous reasons. These include giving more insight into enumerative problems or the study of some properties of graphs. Problems of graph generation may also suggest conjectures or point out counterexamples. The use of generating functions (or functionals) in the enumeration or generation of graphs is standard practice both in mathematics and physics . However, this is by no means obligatory since any method of manipulating graphs may be used.

Furthermore, the problem of generating graphs taking into account their symmetries was considered as early as the 19th century  and more recently for instance, in . In particular, in quantum field theory, generated graphs are weighted by scalars given by the inverses of the orders of their groups of automorphisms . In [7, 8], this was handled for trees and connected multigraphs (with multiple edges and loops allowed), on the level of the symmetric algebra on the vector space of time-ordered field operators. The underlying structure is an algebraic graph representation subsequently developed in . In this representation, graphs are associated with tensors whose indices correspond to the vertex numbers. In the former papers, this made it possible to derive recursion formulas to produce larger graphs from smaller ones by increasing by 1 the number of their vertices or the number of their edges. An interesting property of these formulas is that of satisfying alternative recurrences which relate either a tree or connected multigraph on vertices with all pairs of their connected subgraphs with total number of vertices equal to . In the case of trees, the algorithmic description of the corresponding formula is about the same as that used by Dziobek in his induction proof of Cayley’s formula [10, 11]. Accordingly, the formula induces a recurrence for , that is, the sum of the inverses of the orders of the groups of automorphisms of all the equivalence classes of trees on vertices [12, page 209].

For simplicity, here by graphs we mean simple graphs. However, our results generalize straightforwardly to graphs with multiple edges allowed. One instance of an algorithm for finding the biconnected components of a connected graph is given in . Our goal here is rather to generate all the equivalence classes of connected graphs so that they are decomposed into their biconnected components and have the coefficients announced in the abstract. To this end, we give a suitable graph transformation to produce larger connected graphs from smaller ones by increasing the number of their biconnected components by one unit. This mapping is then used to extend the recurrence of  to connected graphs. This new recurrence decomposes the graphs into their biconnected components and, in addition, can be generalized to restricted classes of connected graphs with specified biconnected components. The proof proceeds as suggested in . That is, given an arbitrary equivalence class whose representative is a graph on edges, say, , we show that every one of the edges of the graph adds to the coefficient of . To this end, we use the fact that labeled vertices are held fixed under any automorphism.

Moreover, in the algebraic representation framework, the result yields a recurrence to generate linear combinations of tensors over the rational numbers. Each tensor represents a connected graph. As required, these linear combinations have the property that the sum of the coefficients of all the tensors representing isomorphic graphs is the inverse of the order of their group of automorphisms. In this context, tensors representing generated graphs are factorized into tensors representing their biconnected components. As in , a key feature of this result is its close relation to the algorithmic description of the computations involved. Indeed, it is easy to read off from this scheme not only algorithms to perform the computations, but even data structures relevant for an implementation.

Furthermore, we prove that when we only consider the restricted class of connected graphs whose biconnected components all have, say, vertices and edges, the corresponding recurrence has an alternative expression which relates connected graphs on biconnected components with all the -tuples of their connected subgraphs with total number of biconnected components equal to . This induces a recurrence for the sum of the inverses of the orders of the groups of automorphisms of all the equivalence classes of connected graphs on biconnected components with that property. The proof uses an identity related to Abel’s binomial theorem [14, 15] and generalizes Dziobek’s induction proof of Cayley’s formula .

This paper is organized as follows. Section 2 reviews the basic concepts of graph theory underlying much of the paper. Section 3 contains the definitions of the elementary graph transformations to be used in the following. Section 4 gives a recursion formula for generating all the equivalence classes of connected graphs in terms of their biconnected components. Sections 5 and 6 review the algebraic representation and some of the linear mappings introduced in [7, 9]. Section 7 derives an algebraic expression for the recurrence of Section 4 and for the particular case in which graphs are such that their biconnected components are all graphs on the same vertex and edge numbers. An alternative formulation for the latter is also given. Finally, Section 8 proves a Cayley-type formula for graphs of that kind.

#### 2. Basics

We briefly review the basic concepts of graph theory that are relevant for the following sections. More details may be found in any standard textbook on graph theory such as .

Let and denote sets. By we denote the set of all the -element subsets of . Also, by we denote the power set of , that is, the set of all the subsets of . By card  we denote the cardinality of the set . Furthermore, we recall that the symmetric difference of the sets and is given by .

Here, a graph is a pair , where is a finite set and . Thus, the elements of are 2-element subsets of . The elements of and are called vertices and edges, respectively. In the following, the vertex set of a graph will often be referred to as , the edge set as . The cardinality of is called the order of , written as . A vertex is said to be incident with an edge if . Then, is an edge at . The two vertices incident with an edge are its endvertices. Moreover, the degree of a vertex is the number of edges at . Two vertices and are said to be adjacent if . If all the vertices of are pairwise adjacent, then is said to be complete. A graph is called a subgraph of a graph if and . A path is a graph on vertices such that ,   for all . The vertices and have degree 1, while the vertices have degree . In this context, the vertices and are linked by and called the endpoint vertices. The vertices are called the inner vertices. A cycle is a graph on vertices such that , for all , every vertex having degree . A graph is said to be connected if every pair of vertices is linked by a path. Otherwise, it is disconnected. Given a graph , a maximal connected subgraph of is called a component of . Furthermore, given a connected graph, a vertex whose removal (together with its incident edges) disconnects the graph is called a cutvertex. A graph that remains connected after erasing any vertex (together with incident edges) (resp. any edge) is said to be 2-connected (resp. 2-edge connected). A -connected graph (resp. -edge connected graph) is also called biconnected (resp. edge-biconnected). Given a connected graph , a biconnected component of is a maximal subset of edges such that the induced subgraph is biconnected (see [17, Section 6.4] for instance). Here, we consider that an isolated vertex is, by convention, a biconnected graph with no biconnected components.

Moreover, given a graph , the set is a vector space over the field such that vector addition is given by the symmetric difference. The cycle space of the graph is defined as the subspace of generated by all the cycles in . The dimension of is called the cyclomatic number of the graph . We recall that , where denotes the number of connected components of the graph .

We now introduce a definition of labeled graph. Let be a finite set. Here, a labeling of a graph is a mapping such that and for all with . In this context, is called a label set, while the graph is said to be labeled with or simply a labeled graph. In the sequel, a labeling of a graph will be referred to as . Moreover, an unlabeled graph is one labeled with the empty set.

Furthermore, an isomorphism between two graphs and is a bijection which satisfies the following conditions: (i) if and only if , (ii). Clearly, an isomorphism defines an equivalence relation on graphs. In particular, an isomorphism of a graph onto itself is called an automorphism (or symmetry) of .

#### 3. Elementary Graph Transformations

We introduce the basic graph transformations to change the number of biconnected components of a connected graph by one unit.

Here, given an arbitrary set , let denote the free vector space on the set over , the set of rational numbers. Also, for all integers and and label sets , let Furthermore, let(i), (ii). In what follows, when we will omit from the upper indices in the previous definitions.

We proceed to the definition of the elementary linear mappings to be used in the following. Note that, for simplicity, our notation does not distinguish between two mappings defined both according to one of the following definitions, one on and the other on with or or . This convention will often be used in the rest of the paper for all the mappings given in this section. Therefore, we will specify the domain of the mappings whenever confusion may arise. (i) Adding a biconnected component to a connected graph: let be a label set. Let be a graph in . Let . For all , let denote the set of biconnected components of such that for all . Let denote the set of all the ordered partitions of the set into disjoint sets: ,  . Furthermore, let denote the set of all the ordered partitions of the set into disjoint sets: . Finally, let be a graph in such that . (In case the graph does not satisfy that property, we consider a graph instead such that and . We will not point this out explicitly in the following.) Let . In this context, for all , define where the graphs satisfy the following:(a) , (b) (c) and for all . The mappings are extended to all of by linearity. For instance, Figure 1 shows the result of applying the mapping to the cutvertex of a -edge connected graph with two biconnected components, where denotes a cycle on four vertices. Furthermore, let . Given a linear combination of graphs , where , we define We proceed to generalize the edge contraction operation given in  to the operation of contracting a biconnected component of a connected graph. (ii) Contracting a biconnected component of a connected graph: let be a label set. Let be a graph in . Let be a biconnected component of , where . Define

where the graph satisfies the following:(a) , where , (b) (c) and . For instance, Figure 2 shows the result of applying the mapping to a -edge connected graph with three biconnected components. We now introduce the following auxiliary mapping.Let be a label set. Let be a graph in . Let . Let be a label set such that . Also, let denote the set of all the ordered partitions of the set into disjoint sets: . In this context, define where the graphs satisfy the following:(a) , (b) , (c) for all and . The mapping is extended to all of by linearity.

#### 4. Generating Connected Graphs

We give a recursion formula to generate all the equivalence classes of connected graphs. The formula depends on the vertex and cyclomatic numbers and produces larger graphs from smaller ones by increasing the number of their biconnected components by one unit. Here, graphs having the same parameters are algebraically represented by linear combinations over coefficients from the rational numbers. The key feature is that the sum of the coefficients of all the graphs in the same equivalence class is given by the inverse of the order of their group of automorphisms. Moreover, the generated graphs are automatically decomposed into their biconnected components.

In the rest of the paper, we often use the following notation: given a group , by we denote the order of . Given a graph , by we denote the group of automorphisms of . Accordingly, given an equivalence class , by we denote the group of automorphisms of all the graphs in . Furthermore, given a set , by we denote the set of equivalence classes of all the graphs in .

We proceed to generalize the recursion formula for generating trees given in  to arbitrary connected graphs.

Theorem 1. For all and suppose that with , is such that for any equivalence class the following holds: (i) there exists such that , (ii) . In this context, given a label set , for all and , define by the following recursion relation: Then, with . Moreover, for any equivalence class , the following holds: (i) there exists such that , (ii) .

Proof. The proof is very analogous to the one given in  (see also [8, 19]).

Lemma 2. Let and be fixed integers. Let be a label set. Let be defined by formula (8). Let denote any equivalence class. Then, there exists such that .

Proof. The proof proceeds by induction on the number of biconnected components . Clearly, the statement is true for . We assume the statement to hold for all the equivalence classes in with and , whose elements have biconnected components. Now, suppose that the elements of have biconnected components. Let with . Recall that by (4) the mappings read as Let denote any graph in . We proceed to show that a graph isomorphic to is generated by applying the mappings to a graph with biconnected components and such that , where is the coefficient of in . Let be any biconnected component of the graph , where . Contracting the graph to the vertex yields a graph with biconnected components. Let denote the equivalence class such that . By the inductive assumption, there exists a graph such that and . Let be the vertex mapped to of by an isomorphism. Relabeling the graph with the empty set yields a graph . Applying the mapping to the graph yields a linear combination of graphs, one of which is isomorphic to . That is, there exists such that .

Lemma 3. Let and be fixed integers. Let be a label set such that . Let be defined by formula (8). Let be an equivalence class such that for all and . Then, .

Proof. The proof proceeds by induction on the number of biconnected components . Clearly, the statement is true for . We assume the statement to hold for all the equivalence classes in with and , whose elements have biconnected components and the property that no vertex is labeled with the empty set. Now, suppose that the elements of have biconnected components. By Lemma 2, there exists a graph such that , where is the coefficient of in . Let . Therefore, . By assumption, for all so that . We proceed to show that . To this end, we check from which graphs with biconnected components, the elements of are generated by the recursion formula (8), and how many times they are generated.
Choose any one of the biconnected components of the graph . Let this be the graph , where . Let so that . Contracting the graph to the vertex with yields a graph with biconnected components. Let denote the equivalence class containing . Since for all , we also have . Let with . By Lemma 2, there exists a graph such that and . By the inductive assumption, . Now, let be the vertex which is mapped to of by an isomorphism. Let be the biconnected graph obtained by relabeling with the empty set. Also, let denote the equivalence class such that . Apply the mapping to the graph . Note that every one of the graphs in the linear combination corresponds to a way of labeling the graph with . Therefore, there are graphs in which are isomorphic to the graph . Since none of the vertices of the graph is labeled with the empty set, the mapping produces a graph isomorphic to from the graph with coefficient . Now, formula (8) prescribes to apply the mappings to the vertex which is mapped to by an isomorphism of every graph in the equivalence class occurring in (with non-zero coefficient). Therefore, where the factor on the right hand side of the first equality is due to the fact that every graph in the equivalence class generates graphs in . Hence, according to formulas (8) and (4), the contribution to is . Distributing this factor between the edges of the graph yields for each edge. Repeating the same argument for every biconnected component of the graph proves that every one of the edges of the graph adds to . Hence, the overall contribution is exactly . This completes the proof.

We now show that satisfies the following property.

Lemma 4. Let and be fixed integers. Let and be label sets such that . Then, .

Proof. Let and be defined as in Section 3. The identity follows by noting that .

Lemma 5. Let and be fixed integers. Let be a label set. Let be defined by formula (8). Let denote any equivalence class. Then, .

Proof. Let be a graph in . If for all , we simply recall Lemma 3. Thus, we may assume that there exists a set such that . Let be a label set such that and . Relabeling the graph with via the mapping such that and for all yields a graph such that . Let be the equivalence class such that . Let with . By Lemma 3, . Suppose now that there are exactly distinct labelings ,   such that , for all , and , where is the graph obtained by relabeling with . Clearly, , where is the coefficient of in . Define for all . Now, repeating the same procedure for every graph in and recalling Lemma 4, we obtain That is, . Since , we obtain .
This completes the proof of Theorem 1.

Figure 3 shows for . Now, given a connected graph , let denote the set of biconnected components of . Given a set , let with the convention . With this notation, Theorem 1 specializes straightforwardly to graphs with specified biconnected components.

Corollary 6. For all and suppose that with and , is such that for any equivalence class , the following holds: (i) there exists such that , (ii) . In this context, for all and , define by the following recursion relation: Then, , where and . Moreover, for any equivalence class the following holds: (i) there exists such that (ii) .

Proof. The result follows from the linearity of the mappings and the fact that larger graphs whose biconnected components are all in can only be produced from smaller ones with the same property.

#### 5. Algebraic Representation of Graphs

We represent graphs by tensors whose indices correspond to the vertex numbers. Our description is essentially that of [7, 9]. From the present section on, we will only consider unlabeled graphs.

Let be a vector space over . Let denote the symmetric algebra on . Then, , where , , and is generated by the free commutative product of elements of . Also, let denote the -fold tensor product of with itself. Recall that the multiplication in is given by the componentwise product: where ,   denote monomials on the elements of for all . We may now proceed to the correspondence between graphs on and some elements of . First, for all with , we define the following tensors in . where is any vector different from zero. (As in Section 3, for simplicity, our notation does not distinguish between elements, say, and with . This convention will often be used in the rest of the paper for all the elements of the algebraic representation. Therefore, we will specify the set containing consider the given elements whenever necessary.) Now, for all with , let (i)a tensor factor in the th position correspond to the vertex of a graph on ,(ii)a tensor correspond to the edge of a graph on .

In this context, given a graph with and , we define the following algebraic representation of graphs.(a)If , then is represented by the tensor . (b)If , then in the graph yields a monomial on the tensors which represent the edges of . More precisely, since for all each tensor represents an edge of the graph , this is uniquely represented by the following tensor given by the componentwise product of the tensors : Figure 4 shows some examples of this correspondence. Furthermore, let be a set of cardinality with . Also, let be a bijection. With this notation, define the following elements of : In terms of graphs, the tensor represents a disconnected graph, say, , on the set consisting of a graph isomorphic to whose vertex set is and isolated vertices in . Figure 5 shows an example.

Furthermore, let denote the tensor algebra on the graded vector space : . In the multiplication is given by concatenation of tensors (e.g., see ): where denote monomials on the elements of for all and . We proceed to generalize the definition of the multiplication to any two positions of the tensor factors. Let . Moreover, define for all . In this context, for all , , we define by the following equation: where (resp. ) means that (resp. ) is excluded from the sequence. In terms of the tensors and the equation previous yields where , if , , if , and if , , if . Clearly, the tensor represents a disconnected graph. Now, let . In , for all , , define the following nonassociative and noncommutative multiplication: The tensor represents the graph on vertices, say, obtained by gluing the vertex of the graph to the vertex of the graph . If both and are connected, the vertex is clearly a cutvertex of the graph . Figure 6 shows an example.

#### 6. Linear Mappings

We recall some of the linear mappings given in .

Let denote the vector space of all the tensors representing biconnected graphs in . Let , where is, by Kirchhoff’s lemma , the maximum cyclomatic number of a biconnected graph on vertices. Let the mapping be given by the following equations: where denotes a biconnected graph on vertices represented by . To define the mappings , we introduce the following bijections: In this context, for all with , the mappings are defined by the following equation: where (resp., ) means that the index (resp., ) is excluded from the sequence. The tensor (resp., ) is constructed from by transferring the monomial on the elements of which occupies the th tensor factor to the th position for all (resp., ). Furthermore, suppose that and that the bijection is such that . In this context, define in agreement with . It is straightforward to verify that the mappings satisfy the following property: where we used the same notation for on the right of either side of the previous equation and as the leftmost operator on the left hand side of the equation. Accordingly, as the leftmost operator on the right hand side of the equation.

Now, for all , define the th iterate of , , recursively as follows: where in formula (28). This can be written in different ways corresponding to the composition of with each of the mappings with . These are all equivalent by formula (26).

Extension to Connected Graphs. We now extend the mappings to the vector space of all the tensors representing connected graphs , where . We proceed to define for .

First, given two sets and , by with we denote the set of elements obtained by applying the mapping to every ordered pair . Also, let and , . In this context, for all and , define as follows: Also, define where denotes the symmetric group on the set and is given by formula (16). Finally, for all , define The elements of are clearly tensors representing connected graphs on biconnected components. By (16) and (21), these may be seen as monomials on tensors representing biconnected graphs with the componentwise product · : so that repeated indices correspond to cutvertices of the associated graphs. In this context, an arbitrary connected graph, say, , on vertices and biconnected components yields Where, for all , , is a biconnected graph on vertices represented by , and is a bijection.

We now extend the mapping to by requiring the mappings to satisfy the following condition: Given a connected graph , the mapping may be thought of as a way of (a) splitting the vertex into two new vertices numbered and and (b) distributing the biconnected components sharing the vertex between the two new ones in all the possible ways. Analogously, the action of the mappings consists of (a) splitting the vertex into new vertices numbered and (b) distributing the biconnected components sharing the vertex between the new ones in all the possible ways.

We now combine the mappings with tensors representing biconnected graphs. Let , and be fixed integers. Let be a bijection. Let be a biconnected graph on vertices represented by the tensor . The tensors (see formula (16)) may be viewed as operators acting on by multiplication. In this context, consider the following mappings given by the composition of with : These are the analog of the mappings given in Section 3. In plain English, the mappings produce a connected graph with vertices from one with vertices in the following way:(a)split the vertex into new vertices, namely, , ,, , (b)distribute the biconnected components containing the split vertex between the new ones in all the possible ways, (c)merge the new vertices into the graph . When the graph is a 2-vertex tree, the mapping coincides with the application of  when acts on by multiplication with a vector.

To illustrate the action of the mappings , consider the graph consisting of two triangles sharing a vertex. Let this be represented by , where denotes a triangle represented by . Let denote a -vertex tree represented by . Applying the mapping to yields Figure 7 shows the linear combination of graphs given by (35) after taking into account that the first and fourth terms as well as the second and third correspond to isomorphic graphs. Note that (resp. ) is the only cutvertex of the graph represented by the first (resp. fourth) term, while and are both cutvertices of the graphs represented by the second or third terms.

#### 7. Further Recursion Relations

Reference  gives two recursion formulas to generate all the equivalence classes of trees with coefficients given by the inverses of the orders of their groups of automorphisms. On the one hand, the main formula is such that larger trees are produced from smaller ones by increasing the number of their biconnected components by one unit. On the other hand, the alternative formula is such that for all , trees on vertices are produced by connecting a vertex of a tree on vertices to a vertex of a tree on vertices in all the possible ways. Theorem 1 generalizes the main formula to connected graphs. It is the aim of this section to derive an alternative formula for a simplified version of the latter.

Let denote a connected graph. Recall the notation introduced in Section 4; denotes the set of biconnected components of and denotes the set of equivalence classes of the graphs in . Given a set , let with the convention . With this notation, in the algebraic setting, Corollary 6 reads as follows.

Theorem 7. For all and , suppose that with and , is such that for any equivalence class the following holds: (i) there exists such that , (ii) . In this context, for all and , define by the following recursion relation: Then, , where and . Moreover, for any equivalence class , the following holds: (i) there exists such that , (ii) .

Now, given a nonempty set , we use the following notation: