An Algebraic Representation of Graphs and Applications to Graph Enumeration

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.


Introduction
As pointed out in [1], 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 [2][3][4].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 [5] and more recently for instance, in [6].In particular, in quantum field theory, generated graphs are weighted by scalars given by the inverses of the orders of their groups of automorphisms [4].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 [9].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  −2 /!, 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 [13].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 [7] 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 [7].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 1/(⋅|aut |) 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 [7][8][9], 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  − 1.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 [10].
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.

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 [16].
Let  and  denote sets.By [] 2 we denote the set of all the 2-element subsets of .Also, by 2  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 Δ := ( ∪ ) \ ( ∩ ).
Here, a graph is a pair  = (, ), where  ⊂ N is a finite set and  ⊆ [] 2 .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  ≥ 2 vertices such that () = {{ 1 ,  2 }, { 2 ,  3 }, . . ., { −1 ,   }},   ∈ () for all  = 1, . . ., .The vertices  1 and   have degree 1, while the vertices  2 , . . .,  −1 have degree 2. In this context, the vertices  1 and   are linked by  and called the endpoint vertices.The vertices  2 , . . .,  −1 are called the inner vertices.A cycle is a graph  on  > 2 vertices such that () = {{ 1 ,  2 }, { 2 ,  3 }, . . ., { −1 ,   }, {  ,  1 }},   ∈ () for all  = 1, . . ., , every vertex having degree 2. 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 2-connected graph (resp.2-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 2 () is a vector space over the field Z 2 such that vector addition is given by the symmetric difference.The cycle space C() of the graph  is defined as the subspace of 2 () generated by all the cycles in .The dimension of C() is called the cyclomatic number of the graph .We recall that dim C() = card() − || + , where  denotes the number of connected components of the graph  [18].
We now introduce a definition of labeled graph.Let  be a finite set.Here, a labeling of a graph  is a mapping  : () → 2  such that ∪ ∈() () =  and () ∩ (  ) = 0 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.

International Journal of Combinatorics 3
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 .

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 Q denote the free vector space on the set  over Q, the set of rational numbers.Also, for all integers  ≥ 1 and  ≥ 0 and label sets , let  ,, = { : || = , dim C () = ,  is labeled by   :  () → 2  } .
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  ,, conn .Let () = {  } =1,..., .For all  = 1, . . ., , let K  denote the set of biconnected components of  such that   ∈ () for all  ∈ K  .Let L denote the set of all the ordered partitions of the set K  into  disjoint sets: L = {  := (K (1)   , . . ., K Furthermore, let J denote the set of all the ordered partitions of the set   (  ) into  disjoint sets: biconn such that () ∩ ( Ĝ) = 0. (In case the graph Ĝ does not satisfy that property, we consider a graph   instead such that   ≅ Ĝ and ()∩(  ) = 0. We will not point this out explicitly in the following.)Let ( Ĝ) = {  } =1,..., .In this context, for all  = 1, . . ., , define where the graphs      satisfy the following: The mappings  Ĝ  are extended to all of Q ,, conn by linearity.For instance, Figure 1 shows the result of applying the mapping We proceed to generalize the edge contraction operation given in [16] 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  ,, conn .Let Ĝ ∈  ,,  biconn be a biconnected component of , where   = ∪ ∈( Ĝ)   ().Define where the graph  * satisfies the following: For instance, Figure 2 shows the result of applying the mapping   4 to a 2-edge connected graph with three biconnected components.We now introduce the following auxiliary mapping.
The mapping    is extended to all of Q ,, by linearity.

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 aut we denote the group of automorphisms of .Accordingly, given an equivalence class A, by autA we denote the group of automorphisms of all the graphs in A. Furthermore, given a set  ⊆  ,, , by E() we denote the set of equivalence classes of all the graphs in .
Proof.The proof proceeds by induction on the number of biconnected components .Clearly, the statement is true for  = 0. We assume the statement to hold for all the equivalence classes in E( −+1,−, conn ) with  = 2, . . .,  − 1 and  = 0, . . ., , whose elements have −1 biconnected components.Now, suppose that the elements of C ∈ E( Let  denote any graph in C. We proceed to show that a graph isomorphic to  is generated by applying the mappings   to the graph  * yields a linear combination of graphs, one of which is isomorphic to .That is, there exists  ≅  such that   > 0. Lemma 3. Let  ≥ 1 and  ≥ 0 be fixed integers.Let  be a label set such that card ≥ .Let  ,,  = ∑ ∈ ,,     be defined by formula (8).Let C ∈ E( ,,  ) be an equivalence class such that   () ̸ = 0 for all  ∈ () and  ∈ C.Then, Proof.The proof proceeds by induction on the number of biconnected components .Clearly, the statement is true for  = 0. We assume the statement to hold for all the equivalence classes in E( −+1,−, conn ) with  = 2, . . .,  − 1 and  = 0, . . ., , whose elements have  − 1 biconnected components and the property that no vertex is labeled with the empty set.Now, suppose that the elements of C ∈ E( ,, conn ) have  biconnected components.By Lemma 2, there exists a graph  ∈ C such that   > 0, where   is the coefficient of  in  ,, conn .Let  := card().Therefore,  =  +  − 1.By assumption,   () ̸ = 0 for all  ∈ () so that |aut C| = 1.We proceed to show that ∑ ∈C   = 1.To this end, we check from which graphs with  − 1 biconnected components, the elements of C are generated by the recursion formula (8), and how many times they are generated.
Choose  to the graph .Note that every one of the graphs in the linear combination     () corresponds to a way of labeling the graph   with   Ĝ() () =   .Therefore, there are |aut A| 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  *  =   > 0. 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 D occurring in  −+1,−, conn (with non-zero coefficient).Therefore, where the factor |aut A| on the right hand side of the first equality is due to the fact that every graph in the equivalence class D generates |aut A| graphs in C. Hence, according to formulas ( 8) and ( 4), the contribution to ∑ ∈C   is   /.Distributing this factor between the   edges of the graph Ĝ yields 1/ 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 1/ to ∑ ∈C   .Hence, the overall contribution is exactly 1.This completes the proof.
We now show that  ,, conn satisfies the following property.
Proof.Let    :  ,, →  ,,∪  and ξ  :  −+1,−, →  −+1,−,∪  be defined as in Section Figure 3: The result of computing all the pairwise non-isomorphic connected graphs as contributions to  , conn via formula (8) up to order  +  ≤ 5.The coefficients in front of graphs are the inverses of the orders of their groups of automorphisms.same procedure for every graph in C and recalling Lemma 4, we obtain That is, ∑ ∈C   = 1/.Since |aut C| = , we obtain This completes the proof of Theorem 1.
Figure 3 shows  , conn for 1 ≤  +  ≤ 5. Now, given a connected graph , let V  denote the set of biconnected components of .Given a set  ⊂ ∪  =2 ∪  =0  , biconn , let  ,  := { ∈  , conn : E(V  ) ⊆ E()} with the convention  1,0  := {({1}, 0)}.With this notation, Theorem 1 specializes straightforwardly to graphs with specified biconnected components.Corollary 6.For all  > 1 and  ≥ 0 suppose that  ,  := ∑   ∈ ,      with    ∈ Q and  , ⊆  ,  , is such that for any equivalence class A ∈ E( , ), the following holds: (i) there exists   ∈ A such that    > 0, (ii) ∑   ∈A    = 1/|aut A|.In this context, for all  ≥ 1 and  ≥ 0, define  ,  ∈ Q ,  by the following recursion relation:   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.

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 Q.Let S() denote the symmetric algebra on .Then, S() = ⨁ ∞ =0 S  (), where S 0 () := Q1, S 1 () = , and S  () is generated by the free commutative product of  elements of .Also, let S() ⊗ denote the -fold tensor product of S() with itself.Recall that the multiplication in S() ⊗ is given by the componentwise product: where   ,    denote monomials on the elements of  for all ,  = 1, . . ., .We may now proceed to the correspondence between graphs on {1, . . ., } and some elements of S() ⊗ .First, for all ,  = 1, . . .,  with  ̸ = , we define the following tensors in S() ⊗ .
where  is any vector different from zero.(As in Section 3, for simplicity, our notation does not distinguish between elements, say,  , ∈ S() ⊗ and  , ∈ S() ⊗  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 ,  = 1, . . .,  with  ̸ = , let (i) a tensor factor in the th position correspond to the vertex  of a graph on {1, . . ., }, (ii) a tensor  , ∈ S() ⊗ correspond to the edge {, } of a graph on {1, . . ., }.

Linear Mappings
We recall some of the linear mappings given in [9].
Extension to Connected Graphs.We now extend the mappings Δ  to the vector space of all the tensors representing connected graphs B * := ⨁ ∞ =0 B ⬦ , where B ⬦0 = Q1.We proceed to define B ⬦ for  > 0.
We now extend the mapping Δ := (1/) ∑  =1 Δ  to B * by requiring the mappings Δ  to satisfy the following condition: Given a connected graph   , the mapping Δ  may be thought of as a way of (a) splitting the vertex  into two new vertices numbered  and  + 1 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 Δ   consists of (a) splitting the vertex  into  + 1 new vertices numbered ,  + 1, . . .,  +  and (b) distributing the biconnected components sharing the vertex  between the  + 1 new ones in all the possible ways.
To illustrate the action of the mappings     (1),...,  () ∘ Δ  , consider the graph  consisting of two triangles sharing a vertex.Let this be represented by    graphs.Note that 3 (resp.4) is the only cutvertex of the graph represented by the first (resp.fourth) term, while 3 and 4 are both cutvertices of the graphs represented by the second or third terms.

Further Recursion Relations
Reference [7] 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  ≥ 2, 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.
For  = 2 and  = 0, we recover the formula to generate trees of [7].As in that paper and [8,23], we may extend the result to obtain further interesting recursion relations.

𝐶 4 𝑖
to the cutvertex of a 2-edge connected graph with two biconnected components, where  4 denotes a cycle on four vertices.Furthermore, let  ⊆  , biconn .Given a linear combination of graphs  = ∑ ∈   , where   ∈ Q, we define

Figure 1 : 4 𝑖Figure 2 :
Figure 1: The linear combination of graphs obtained by applying the mapping   4  to the cutvertex of the graph consisting of two triangles sharing one vertex.
(a) split the vertex  into  new vertices, namely, , +1,. ..,  +  − 1, (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 .

Figure 7 Figure 7 :
Figure7shows 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