An Algebraic Representation of Graphs and Applications to Graph Enumeration

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 [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 [2–4]. 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 𝑛 vertices with all pairs of their connected subgraphs with total number of vertices equal to 𝑛. 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 𝑛^𝑛−2/𝑛!, that is, the sum of the inverses of the orders of the groups of automorphisms of all the equivalence classes of trees on 𝑛 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 𝑚 edges, say, 𝐺, we show that every one of the 𝑚 edges of the graph 𝐺 adds 1/(𝑚⋅|aut𝐺|) to the coefficient of 𝐺. 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 [7–9], 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, 𝑝 vertices and 𝑟 edges, the corresponding recurrence has an alternative expression which relates connected graphs on 𝜇 biconnected components with all the 𝑝-tuples of their connected subgraphs with total number of biconnected components equal to 𝜇−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 𝜇 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.

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 [16].

Let 𝐴 and 𝐵 denote sets. By [𝐴]² we denote the set of all the 2-element subsets of 𝐴. Also, by 2^𝐴 we denote the power set of 𝐴, that is, the set of all the subsets of 𝐴. By card 𝐴 we denote the cardinality of the set 𝐴. Furthermore, we recall that the symmetric difference of the sets 𝐴 and 𝐵 is given by 𝐴Δ𝐵∶=(𝐴∪𝐵)⧵(𝐴∩𝐵).

Here, a graph is a pair 𝐺=(𝑉,𝐸), where 𝑉⊂ℕ is a finite set and 𝐸⊆[𝑉]². Thus, the elements of 𝐸 are 2-element subsets of 𝑉. The elements of 𝑉 and 𝐸 are called vertices and edges, respectively. In the following, the vertex set of a graph 𝐺 will often be referred to as 𝑉(𝐺), the edge set as 𝐸(𝐺). The cardinality of 𝑉(𝐺) is called the order of 𝐺, written as |𝐺|. A vertex 𝑣 is said to be incident with an edge 𝑒 if 𝑣∈𝑒. Then, 𝑒 is an edge at 𝑣. The two vertices incident with an edge are its endvertices. Moreover, the degree of a vertex 𝑣 is the number of edges at 𝑣. Two vertices 𝑣 and 𝑢 are said to be adjacent if {𝑣,𝑢}∈𝐸. If all the vertices of 𝐺 are pairwise adjacent, then 𝐺 is said to be complete. A graph 𝐺^∗ is called a subgraph of a graph 𝐺 if 𝑉(𝐺^∗)⊆𝑉(𝐺) and 𝐸(𝐺^∗)⊆𝐸(𝐺). A path is a graph 𝑃 on 𝑛≥2 vertices such that 𝐸(𝑃)={{𝑣₁,𝑣₂},{𝑣₂,𝑣₃},…,{𝑣_𝑛−1,𝑣_𝑛}},  𝑣_𝑗∈𝑉(𝑃) for all 𝑗=1,…,𝑛. The vertices 𝑣₁ and 𝑣_𝑛 have degree 1, while the vertices 𝑣₂,…,𝑣_𝑛−1 have degree 2. In this context, the vertices 𝑣₁ and 𝑣_𝑛 are linked by 𝑃 and called the endpoint vertices. The vertices 𝑣₂,…,𝑣_𝑛−1 are called the inner vertices. A cycle is a graph 𝐶 on 𝑛>2 vertices such that 𝐸(𝐶)={{𝑣₁,𝑣₂},{𝑣₂,𝑣₃},…,{𝑣_𝑛−1,𝑣_𝑛},{𝑣_𝑛,𝑣₁}}, 𝑣_𝑗∈𝑉(𝐶) for all 𝑗=1,…,𝑛, 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 𝐺, a maximal connected subgraph of 𝐺 is called a component of 𝐺. 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 𝐻, a biconnected component of 𝐻 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 𝐺, the set 2^𝐸(𝐺) is a vector space over the field ℤ₂ such that vector addition is given by the symmetric difference. The cycle space 𝒞(𝐺) of the graph 𝐺 is defined as the subspace of 2^𝐸(𝐺) generated by all the cycles in 𝐺. The dimension of 𝒞(𝐺) is called the cyclomatic number of the graph 𝐺. We recall that dim𝒞(𝐺)=card𝐸(𝐺)−|𝐺|+𝑐, where 𝑐 denotes the number of connected components of the graph 𝐺 [18].

We now introduce a definition of labeled graph. Let 𝐿 be a finite set. Here, a labeling of a graph 𝐺 is a mapping 𝑙∶𝑉(𝐺)→2^𝐿 such that ∪_{𝑣∈𝑉(𝐺)}𝑙(𝑣)=𝐿 and 𝑙(𝑣)∩𝑙(𝑣^′)=∅ for all 𝑣,𝑣^′∈𝑉(𝐺) with 𝑣≠𝑣^′. In this context, 𝐿 is called a label set, while the graph 𝐺 is said to be labeled with 𝐿 or simply a labeled graph. In the sequel, a labeling of a graph 𝐺 will be referred to as 𝑙_𝐺. Moreover, an unlabeled graph is one labeled with the empty set.

Furthermore, an isomorphism between two graphs 𝐺 and 𝐺^∗ is a bijection 𝜑∶𝑉(𝐺)→𝑉(𝐺^∗) which satisfies the following conditions: (i){𝑣,𝑣^′}∈𝐸(𝐺) if and only if {𝜑(𝑣),𝜑(𝑣^′)}∈𝐸(𝐺^∗), (ii)𝐿∩𝑙_𝐺(𝑣)=𝐿∩𝑙_𝐺^∗(𝜑(𝑣)). Clearly, an isomorphism defines an equivalence relation on graphs. In particular, an isomorphism of a graph 𝐺 onto itself is called an automorphism (or symmetry) of 𝐺.

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 𝑋, let ℚ𝑋 denote the free vector space on the set 𝑋 over ℚ, the set of rational numbers. Also, for all integers 𝑛≥1 and 𝑘≥0 and label sets 𝐿, let 𝑉^{𝑛,𝑘,𝐿}=𝐺∶|𝐺|=𝑛,dim𝒞(𝐺)=𝑘,𝐺islabeledby𝑙_𝐺∶𝑉(𝐺)→2^𝐿. (1) Furthermore, let(i)𝑉^{𝑛,𝑘,𝐿}_conn={𝐺∈𝑉^{𝑛,𝑘,𝐿}∶𝐺isconnected}, (ii)𝑉^{𝑛,𝑘,𝐿}_biconn={𝐺∈𝑉^{𝑛,𝑘,𝐿}∶𝐺isbiconnected}. In what follows, when 𝐿=∅ we will omit 𝐿 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 𝑉^{𝑛,𝑘,𝐿} and the other on 𝑉^{𝑝,𝑞,𝐿′} with 𝑛≠𝑝 or 𝑘≠𝑞 or 𝐿≠𝐿^′. 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 𝐿 be a label set. Let 𝐺 be a graph in 𝑉^{𝑛,𝑘,𝐿}_conn. Let 𝑉(𝐺)={𝑣_𝑖}_{𝑖=1,…,𝑛}. For all 𝑖=1,…,𝑛, let 𝒦_𝑖 denote the set of biconnected components of 𝐺 such that 𝑣_𝑖∈𝑉(𝐻) for all 𝐻∈𝒦_𝑖. Let ℒ denote the set of all the ordered partitions of the set 𝒦_𝑖 into 𝑝 disjoint sets: ℒ={𝑧_𝑖∶=(𝒦_𝑖⁽¹⁾,…,𝒦_𝑖^(𝑝))∶∪^𝑝_𝑙=1𝒦_𝑖^(𝑙)=𝒦_𝑖and𝒦_𝑖^(𝑙)∩𝒦^(𝑙′)_𝑖=∅∀𝑙,𝑙^′=1,…,  𝑝with𝑙≠𝑙^′}. Furthermore, let 𝒥 denote the set of all the ordered partitions of the set 𝑙_𝐺(𝑣_𝑖) into 𝑝 disjoint sets: 𝒥={𝑤_𝑖∶=(𝑙_𝐺(𝑣_𝑖)⁽¹⁾,…,𝑙_𝐺(𝑣_𝑖)^(𝑝))∶∪^𝑝_𝑙=1𝑙_𝐺(𝑣_𝑖)^(𝑙)=𝑙_𝐺(𝑣_𝑖)and𝑙_𝐺(𝑣_𝑖)^(𝑙)∩𝑙_𝐺(𝑣_𝑖)^(𝑙′)=∅∀𝑙,𝑙^′=1,…,𝑝with𝑙≠𝑙^′}. Finally, let 𝐺 be a graph in 𝑉^𝑝,𝑞_biconn such that 𝑉(𝐺)∩𝑉(𝐺)=∅. (In case the graph 𝐺 does not satisfy that property, we consider a graph 𝐺^′ instead such that 𝐺^′≅𝐺 and 𝑉(𝐺)∩𝑉(𝐺^′)=∅. We will not point this out explicitly in the following.) Let 𝑉(𝐺)={𝑢_𝑙}_{𝑙=1,…,𝑝}. In this context, for all 𝑖=1,…,𝑛, define 𝑟𝐺_𝑖∶ℚ𝑉^{𝑛,𝑘,𝐿}_conn→ℚ𝑉^{𝑛+𝑝−1,𝑘+𝑞,𝐿}_conn𝐺↦_{𝑧𝑖∈ℒ;𝑤𝑖∈𝒥}𝐺^𝑧𝑖_𝑤𝑖, (2) where the graphs 𝐺^𝑧𝑖_𝑤𝑖 satisfy the following:(a) 𝑉(𝐺^𝑧𝑖_𝑤𝑖)=𝑉(𝐺)⧵{𝑣_𝑖}∪𝑉(𝐺), (b) 𝐸(𝐺^𝑧𝑖_𝑤𝑖)=𝐸(𝐺)∪{{𝑥,𝑦}∈𝐸(𝐺)∶𝑣_𝑖∉{𝑥,𝑦}}^𝑝_𝑙=1𝑥,𝑢_𝑙∶𝑥,𝑣_𝑖∈𝐸(𝐺),𝑥∈∪_{𝐻∗∈𝒦𝑖}^(𝑙)𝑉𝐻^∗, (3)(c) 𝑙_𝐺^𝑧_𝑖𝑤𝑖|_{𝑉(𝐺)⧵{𝑣𝑖}}=𝑙_𝐺|_{𝑉(𝐺)⧵{𝑣𝑖}} and 𝑙_𝐺^𝑧_𝑖𝑤𝑖(𝑢_𝑙)=𝑙_𝐺(𝑣_𝑖)^(𝑙) for all 𝑙=1,…,𝑝.  The mappings 𝑟𝐺_𝑖 are extended to all of ℚ𝑉^{𝑛,𝑘,𝐿}_conn by linearity. For instance, Figure 1 shows the result of applying the mapping 𝑟^𝐶4_𝑖 to the cutvertex of a 2-edge connected graph with two biconnected components, where 𝐶₄ denotes a cycle on four vertices.  Furthermore, let 𝑋⊆𝑉^𝑝,𝑞_biconn. Given a linear combination of graphs 𝜗=∑_𝐺∈𝑋𝛼_𝐺𝐺, where 𝛼_𝐺∈ℚ, we define 𝑟^𝜗_𝑖∶=_𝐺∈𝑋𝛼_𝐺𝑟^𝐺_𝑖. (4) 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 𝐿 be a label set. Let 𝐺 be a graph in 𝑉^{𝑛,𝑘,𝐿}_conn. Let 𝐺∈𝑉^{𝑝,𝑞,𝐿′}_biconn be a biconnected component of 𝐺, where 𝐿^′=∪_𝑣∈𝑉(𝐺₎𝑙_𝐺(𝑣). Define 𝑐𝐺∶𝑉^{𝑛,𝑘,𝐿}_conn→𝑉^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn;𝐺↦𝐺^∗, (5)

Figure 1 

The linear combination of graphs obtained by applying the mapping 𝑟𝐶4𝑖 to the cutvertex of the graph consisting of two triangles sharing one vertex.

 where the graph 𝐺^∗ satisfies the following:(a) 𝑉(𝐺^∗)=𝑉(𝐺)⧵𝑉(𝐺)∪{𝑣}, where 𝑣∶=min{𝑣^′∈ℕ∶𝑣^′∉𝑉(𝐺)⧵𝑉(𝐺)}, (b) 𝐸(𝐺^∗)={{𝑥,𝑦}∈𝐸(𝐺)∶{𝑥,𝑦}∩𝐸(𝐺)=∅}∪{𝑥,𝑣}∶𝑥,𝑦∈𝐸(𝐺)⧵𝐸𝐺and𝑦∈𝑉𝐺, (6)(c) 𝑙_𝐺^∗|_{𝑉(𝐺)⧵𝑉(}𝐺₎=𝑙_𝐺|_{𝑉(𝐺)⧵𝑉(}𝐺₎ and 𝑙_𝐺^∗(𝑣)=∪_𝑢∈𝑉(𝐺₎𝑙_𝐺(𝑢).  For instance, Figure 2 shows the result of applying the mapping 𝑐_𝐶4 to a 2-edge connected graph with three biconnected components.  We now introduce the following auxiliary mapping. Let 𝐿 be a label set. Let 𝐺 be a graph in 𝑉^{𝑛,𝑘,𝐿}. Let 𝑉(𝐺)={𝑣_𝑖}_{𝑖=1,…,𝑛}. Let 𝐿^′ be a label set such that 𝐿∩𝐿^′=∅. Also, let ℐ denote the set of all the ordered partitions of the set 𝐿^′ into 𝑛 disjoint sets: ℐ={𝑦∶=(𝐿^′(1),…,𝐿^′(𝑛))∶∪^𝑛_𝑗=1𝐿^′(𝑗)=𝐿^′and𝐿^′(𝑖)∩𝐿^′(𝑗)=∅∀𝑖,𝑗=1,…,𝑛with𝑖≠𝑗}. In this context, define 𝜉_𝐿′∶ℚ𝑉^{𝑛,𝑘,𝐿}→ℚ𝑉^{𝑛,𝑘,𝐿∪𝐿′};𝐺↦_𝑦∈ℐ𝐺_𝑦, (7) where the graphs 𝐺_𝑦 satisfy the following:(a) 𝑉(𝐺_𝑦)=𝑉(𝐺), (b) 𝐸(𝐺_𝑦)=𝐸(𝐺), (c) 𝑙_𝐺𝑦(𝑣_𝑖)=𝑙_𝐺(𝑣_𝑖)∪𝐿^′(𝑖) for all 𝑖=1,…,𝑛 and 𝑣_𝑖∈𝑉(𝐺).  The mapping 𝜉_𝐿′ is extended to all of ℚ𝑉^{𝑛,𝑘,𝐿} by linearity.

Figure 2 

The graph obtained by applying the mapping 𝑐𝐶4 to a graph with three biconnected components.

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 𝐻, by |𝐻| we denote the order of 𝐻. Given a graph 𝐺, by aut𝐺 we denote the group of automorphisms of 𝐺. Accordingly, given an equivalence class 𝒜, by aut𝒜 we denote the group of automorphisms of all the graphs in 𝒜. Furthermore, given a set 𝑊⊆𝑉^{𝑛,𝑘,𝐿}, by ℰ(𝑊) we denote the set of equivalence classes of all the graphs in 𝑊.

We proceed to generalize the recursion formula for generating trees given in [7] to arbitrary connected graphs.

Theorem 1. For all 𝑝>1 and 𝑞≥0 suppose that 𝛽^𝑝,𝑞_{𝑏𝑖𝑐𝑜𝑛𝑛}∶=∑_{𝐺′∈𝑉}^𝑝,𝑞_{𝑏𝑖𝑐𝑜𝑛𝑛}𝜎_𝐺′𝐺^′ with 𝜎_𝐺′∈ℚ, is such that for any equivalence class 𝒜∈ℰ(𝑉^𝑝,𝑞_{𝑏𝑖𝑐𝑜𝑛𝑛}) the following holds: (i) there exists 𝐺^′∈𝒜 such that 𝜎_𝐺′>0, (ii) ∑_{𝐺′∈𝒜}𝜎_𝐺′=1/|𝑎𝑢𝑡𝒜|. In this context, given a label set 𝐿, for all 𝑛≥1 and 𝑘≥0, define 𝛽^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}∈ℚ𝑉^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛} by the following recursion relation: 𝛽^1,0,𝐿_{𝑐𝑜𝑛𝑛}∶=𝐺,𝑤ℎ𝑒𝑟𝑒𝐺=({1},∅),𝑙_𝐺(1)𝛽=𝐿,_{1,𝑘,𝐿𝑐𝑜𝑛𝑛}𝛽∶=0𝑖𝑓𝑘>0,_{𝑛,𝑘,𝐿𝑐𝑜𝑛𝑛}1∶=×𝑘+𝑛−1_{𝑘𝑛𝑞=0𝑝=2𝑛−𝑝+1𝑖=1}𝑞+𝑝−1𝑟^{𝛽𝑝,𝑞𝑏𝑖𝑐𝑜𝑛𝑛}_𝑖𝛽^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_{𝑐𝑜𝑛𝑛}. (8) Then, 𝛽^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}=∑_𝐺∈𝑉^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}𝛼_𝐺𝐺 with 𝛼_𝐺∈ℚ. Moreover, for any equivalence class 𝒞∈ℰ(𝑉^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}), the following holds: (i) there exists 𝐺∈𝒞 such that 𝛼_𝐺>0, (ii) ∑_𝐺∈𝒞𝛼_𝐺=1/|𝑎𝑢𝑡𝒞|.

Proof. The proof is very analogous to the one given in [7] (see also [8, 19]).

Lemma 2. Let 𝑛≥1 and 𝑘≥0 be fixed integers. Let 𝐿 be a label set. Let 𝛽^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}=∑_𝐺∈𝑉^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}𝛼_𝐺𝐺 be defined by formula (8). Let 𝒞∈ℰ(𝑉^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}) denote any equivalence class. Then, there exists 𝐺∈𝒞 such that 𝛼_𝐺>0.

Proof. The proof proceeds by induction on the number of biconnected components 𝜇. Clearly, the statement is true for 𝜇=0. We assume the statement to hold for all the equivalence classes in ℰ(𝑉^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn) with 𝑝=2,…,𝑛−1 and 𝑞=0,…,𝑘, whose elements have 𝜇−1 biconnected components. Now, suppose that the elements of 𝒞∈ℰ(𝑉^{𝑛,𝑘,𝐿}_conn) have 𝜇 biconnected components. Let 𝛽^𝑝,𝑞_biconn=∑_{𝐺′∈𝑉}^𝑝,𝑞_biconn𝜎_𝐺′𝐺^′ with 𝜎_𝐺′∈ℚ. Recall that by (4) the mappings 𝑟^{𝛽𝑝,𝑞biconn}_𝑖 read as 𝑟^{𝛽𝑝,𝑞biconn}_𝑖∶=_{𝐺′∈𝑉𝑝,𝑞biconn}𝜎_𝐺′𝑟^𝐺′_𝑖. (9) Let 𝐺 denote any graph in 𝒞. We proceed to show that a graph isomorphic to 𝐺 is generated by applying the mappings 𝑟^{𝛽𝑝,𝑞biconn}_𝑖 to a graph 𝐺^∗∈𝑉^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn with 𝜇−1 biconnected components and such that 𝜈_𝐺^∗>0, where 𝜈_𝐺^∗ is the coefficient of 𝐺^∗ in 𝛽^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn. Let 𝐺∈𝑉^{𝑝,𝑞,𝐿′}_biconn be any biconnected component of the graph 𝐺, where 𝐿^′=∪_𝑣∈𝑉(𝐺₎𝑙_𝐺(𝑣). Contracting the graph 𝐺 to the vertex 𝑢∶=min{𝑢^′∈ℕ∶𝑢^′∉𝑉(𝐺)⧵𝑉(𝐺)} yields a graph 𝑐𝐺(𝐺)∈𝑉^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn with 𝜇−1 biconnected components. Let 𝒟∈ℰ(𝑉^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn) denote the equivalence class such that 𝑐𝐺(𝐺)∈𝒟. By the inductive assumption, there exists a graph 𝐻^∗∈𝒟 such that 𝐻^∗𝐺≅𝑐(𝐺) and 𝜈_𝐻^∗>0. Let 𝑣_𝑗∈𝑉(𝐻^∗) be the vertex mapped to 𝑢 of 𝑐𝐺(𝐺) by an isomorphism. Relabeling the graph 𝐺 with the empty set yields a graph 𝐺^′∈𝑉^𝑝,𝑞_biconn. Applying the mapping 𝑟^𝐺′_𝑗 to the graph 𝐻^∗ yields a linear combination of graphs, one of which is isomorphic to 𝐺. That is, there exists 𝐻≅𝐺 such that 𝛼_𝐻>0.

Lemma 3. Let 𝑛≥1 and 𝑘≥0 be fixed integers. Let 𝐿 be a label set such that 𝑐𝑎𝑟𝑑𝐿≥𝑛. Let 𝛽^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}=∑_𝐺∈𝑉^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}𝛼_𝐺𝐺 be defined by formula (8). Let 𝒞∈ℰ(𝑉^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}) be an equivalence class such that 𝑙_𝐺(𝑣)≠∅ for all 𝑣∈𝑉(𝐺) and 𝐺∈𝒞. Then, ∑_𝐺∈𝒞𝛼_𝐺=1.

Proof. The proof proceeds by induction on the number of biconnected components 𝜇. Clearly, the statement is true for 𝜇=0. We assume the statement to hold for all the equivalence classes in ℰ(𝑉^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn) with 𝑝=2,…,𝑛−1 and 𝑞=0,…,𝑘, whose elements have 𝜇−1 biconnected components and the property that no vertex is labeled with the empty set. Now, suppose that the elements of 𝒞∈ℰ(𝑉^{𝑛,𝑘,𝐿}_conn) have 𝜇 biconnected components. By Lemma 2, there exists a graph 𝐺∈𝒞 such that 𝛼_𝐺>0, where 𝛼_𝐺 is the coefficient of 𝐺 in 𝛽^{𝑛,𝑘,𝐿}_conn. Let 𝑚∶=card𝐸(𝐺). Therefore, 𝑚=𝑘+𝑛−1. By assumption, 𝑙_𝐺(𝑣)≠∅ for all 𝑣∈𝑉(𝐺) so that |aut𝒞|=1. We proceed to show that ∑_𝐺∈𝒞𝛼_𝐺=1. To this end, we check from which graphs with 𝜇−1 biconnected components, the elements of 𝒞 are generated by the recursion formula (8), and how many times they are generated.
Choose any one of the 𝜇 biconnected components of the graph 𝐺∈𝒞. Let this be the graph 𝐺∈𝑉^{𝑝,𝑞,𝐿′}_biconn, where 𝐿^′=∪_𝑣∈𝑉(𝐺₎𝑙_𝐺(𝑣). Let 𝑚^′∶=card𝐸(𝐺) so that 𝑚^′=𝑞+𝑝−1. Contracting the graph 𝐺 to the vertex 𝑢 with 𝑢∶=min{𝑢^′∈ℕ∶𝑢^′∉𝑉(𝐺)⧵𝑉(𝐺)} yields a graph 𝑐𝐺(𝐺)∈𝑉^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn with 𝜇−1 biconnected components. Let 𝒟∈ℰ(𝑉^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn) denote the equivalence class containing 𝑐𝐺(𝐺). Since 𝑙_𝑐𝐺^(𝐺)(𝑣)≠∅ for all 𝐺𝑣∈𝑉(𝑐(𝐺)), we also have |aut𝒟|=1. Let 𝛽^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn=∑_{𝐺∗∈𝑉}^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn𝜈_𝐺^∗𝐺^∗ with 𝜈_𝐺^∗∈ℚ. By Lemma 2, there exists a graph 𝐻∈𝒟 such that 𝐺𝐻≅𝑐(𝐺) and 𝜈_𝐻>0. By the inductive assumption, ∑_𝐺^∗_∈𝒟𝜈_𝐺^∗=1. Now, let 𝑣_𝑗∈𝑉(𝐻) be the vertex which is mapped to 𝑢 of 𝑐𝐺(𝐺) by an isomorphism. Let 𝐺^′∈𝑉^𝑝,𝑞_biconn be the biconnected graph obtained by relabeling 𝐺 with the empty set. Also, let 𝒜∈ℰ(𝑉^𝑝,𝑞_biconn) denote the equivalence class such that 𝐺^′∈𝒜. Apply the mapping 𝑟^𝐺′_𝑗 to the graph 𝐻. Note that every one of the graphs in the linear combination 𝑟^𝐺′_𝑗(𝐻) corresponds to a way of labeling the graph 𝐺^′ with 𝑙_𝑐𝐺^(𝐺)(𝑢)=𝐿^′. Therefore, there are |aut𝒜| graphs in 𝑟^𝐺′_𝑗(𝐻) which are isomorphic to the graph 𝐺. Since none of the vertices of the graph 𝐻 is labeled with the empty set, the mapping 𝑟^𝐺′_𝑗 produces a graph isomorphic to 𝐺 from the graph 𝐻 with coefficient 𝛼^∗_𝐺=𝜈_𝐻>0. Now, formula (8) prescribes to apply the mappings 𝑟^𝐺′_𝑖 to the vertex which is mapped to 𝑢 by an isomorphism of every graph in the equivalence class 𝒟 occurring in 𝛽^{𝑛−𝑝+1,𝑘−𝑞,𝐿}_conn (with non-zero coefficient). Therefore, _𝐺∈𝒞𝛼^∗_𝐺=|aut𝒜|⋅_{𝐺∗∈𝒟}𝜈_𝐺^∗=|aut𝒜|, (10) where the factor |aut𝒜| on the right hand side of the first equality is due to the fact that every graph in the equivalence class 𝒟 generates |aut𝒜| graphs in 𝒞. Hence, according to formulas (8) and (4), the contribution to ∑_𝐺∈𝒞𝛼_𝐺 is 𝑚^′/𝑚. Distributing this factor between the 𝑚^′ edges of the graph 𝐺 yields 1/𝑚 for each edge. Repeating the same argument for every biconnected component of the graph 𝐺 proves that every one of the 𝑚 edges of the graph 𝐺 adds 1/𝑚 to ∑_𝐺∈𝒞𝛼_𝐺. Hence, the overall contribution is exactly 1. This completes the proof.

We now show that 𝛽^{𝑛,𝑘,𝐿}_conn satisfies the following property.

Lemma 4. Let 𝑛≥1 and 𝑘≥0 be fixed integers. Let 𝐿 and 𝐿^′ be label sets such that 𝐿∩𝐿^′=∅. Then, 𝛽^{𝑛,𝑘,𝐿∪𝐿′}_{𝑐𝑜𝑛𝑛}=𝜉_𝐿′(𝛽^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}).

Proof. Let 𝜉_𝐿′∶𝑉^{𝑛,𝑘,𝐿}→𝑉^{𝑛,𝑘,𝐿∪𝐿′} and 𝜉̃_𝐿′∶𝑉^{𝑛−𝑝+1,𝑘−𝑞,𝐿}→𝑉^{𝑛−𝑝+1,𝑘−𝑞,𝐿∪𝐿′} be defined as in Section 3. The identity follows by noting that 𝜉_𝐿′∘𝑟^{𝛽𝑝,𝑞biconn}_𝑖=𝑟^{𝛽𝑝,𝑞biconn}_𝑖∘𝜉̃_𝐿′.

Lemma 5. Let 𝑛≥1 and 𝑘≥0 be fixed integers. Let 𝐿 be a label set. Let 𝛽^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}=∑_𝐺∈𝑉^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}𝛼_𝐺𝐺 be defined by formula (8). Let 𝒞∈ℰ(𝑉^{𝑛,𝑘,𝐿}_{𝑐𝑜𝑛𝑛}) denote any equivalence class. Then, ∑_𝐺∈𝒞𝛼_𝐺=1/|𝑎𝑢𝑡(𝒞)|.

Proof. Let 𝐺 be a graph in 𝒞. If 𝑙_𝐺(𝑣)≠∅ for all 𝑣∈𝑉(𝐺), we simply recall Lemma 3. Thus, we may assume that there exists a set 𝑉^′⊆𝑉(𝐺) such that 𝑙_𝐺(𝑉^′)={∅}. Let 𝐿^′ be a label set such that 𝐿∩𝐿^′=∅ and card𝐿^′=card𝑉^′. Relabeling the graph 𝐺 with 𝐿∪𝐿^′ via the mapping 𝑙∶𝑉(𝐺)→2^{𝐿∪𝐿′} such that 𝑙|_{𝑉(𝐺)⧵𝑉′}=𝑙_𝐺|_{𝑉(𝐺)⧵𝑉′} and 𝑙(𝑣)≠∅ for all 𝑣∈𝑉^′ yields a graph 𝐺^′∈𝑉^{𝑛,𝑘,𝐿∪𝐿′}_conn such that |aut𝐺^′|=1. Let 𝒟∈ℰ(𝑉^{𝑛,𝑘,𝐿∪𝐿′}_conn) be the equivalence class such that 𝐺^′∈𝒟. Let 𝛽^{𝑛,𝑘,𝐿∪𝐿′}_conn=∑_{𝐺′∈𝑉′𝑛,𝑘,𝐿∪𝐿conn}𝜈_𝐺′𝐺^′ with 𝜈_𝐺′∈ℚ. By Lemma 3, ∑_{𝐺′∈𝒟}𝜈_𝐺′=1. Suppose now that there are exactly 𝑇 distinct labelings 𝑙_𝑗∶𝑉(𝐺)→2^{𝐿∪𝐿′},  𝑗=1,…,𝑇 such that 𝑙_𝑗|_{𝑉(𝐺)⧵𝑉′}=𝑙_𝐺|_{𝑉(𝐺)⧵𝑉′}, 𝑙_𝑗(𝑣)≠∅ for all 𝑣∈𝑉^′, and 𝐺_𝑗∈𝒟, where 𝐺_𝑗 is the graph obtained by relabeling 𝐺 with 𝑙_𝑗. Clearly, 𝜈_𝐺𝑗=𝛼_𝐺>0, where 𝜈_𝐺𝑗 is the coefficient of 𝐺_𝑗 in 𝛽^{𝑛,𝑘,𝐿∪𝐿′}_conn. Define 𝑓_𝑗∶𝐺↦𝐺_𝑗 for all 𝑗=1,…,𝑇. Now, repeating the same procedure for every graph in 𝒞 and recalling Lemma 4, we obtain _{𝐺′∈𝒟}𝜈_𝐺′=^𝑇_{𝑗=1𝐺∈𝒞}𝜈_{𝑓𝑗(𝐺)}=𝑇_𝐺∈𝒞𝛼_𝐺=1. (11) That is, ∑_𝐺∈𝒞𝛼_𝐺=1/𝑇. Since |aut𝒞|=𝑇, we obtain ∑_𝐺∈𝒞𝛼_𝐺=1/|aut(𝒞)|.
This completes the proof of Theorem 1.

Figure 3 shows 𝛽^𝑛,𝑘_conn for 1≤𝑛+𝑘≤5. Now, given a connected graph 𝐺, let 𝒱_𝐺 denote the set of biconnected components of 𝐺. Given a set 𝑋⊂∪^𝑛_𝑝=2∪^𝑘_𝑞=0𝑉^𝑝,𝑞_biconn, let 𝑉_𝑋^𝑛,𝑘∶={𝐺∈𝑉^𝑛,𝑘_conn∶ℰ(𝒱_𝐺)⊆ℰ(𝑋)} with the convention 𝑉_𝑋^1,0∶={({1},∅)}. With this notation, Theorem 1 specializes straightforwardly to graphs with specified biconnected components.

Figure 3 

The result of computing all the pairwise non-isomorphic connected graphs as contributions to 𝛽𝑛,𝑘conn via formula (8) up to order 𝑛+𝑘≤5. The coefficients in front of graphs are the inverses of the orders of their groups of automorphisms.

Corollary 6. For all 𝑝>1 and 𝑞≥0 suppose that 𝛾^𝑝,𝑞_{𝑏𝑖𝑐𝑜𝑛𝑛}∶=∑_{𝐺′∈𝑋𝑝,𝑞}𝜎_𝐺′𝐺^′ with 𝜎_𝐺′∈ℚ and 𝑋^𝑝,𝑞⊆𝑉^𝑝,𝑞_{𝑏𝑖𝑐𝑜𝑛𝑛}, is such that for any equivalence class 𝒜∈ℰ(𝑋^𝑝,𝑞), the following holds: (i) there exists 𝐺^′∈𝒜 such that 𝜎_𝐺′>0, (ii) ∑_{𝐺′∈𝒜}𝜎_𝐺′=1/|𝑎𝑢𝑡𝒜|. In this context, for all 𝑛≥1 and 𝑘≥0, define 𝛾^𝑛,𝑘_{𝑐𝑜𝑛𝑛}∈ℚ𝑉^𝑛,𝑘_{𝑐𝑜𝑛𝑛} by the following recursion relation: 𝛾^1,0_{𝑐𝑜𝑛𝑛}1}),𝛾∶=𝐺,𝑤ℎ𝑒𝑟𝑒𝐺=({,∅_{1,𝑘𝑐𝑜𝑛𝑛}𝛾∶=0if𝑘>0,_{𝑛,𝑘𝑐𝑜𝑛𝑛}1∶=×𝑘+𝑛−1_{𝑘𝑛𝑞=0𝑝=2𝑛−𝑝+1𝑖=1}𝑞+𝑝−1𝑟^{𝛾𝑝,𝑞𝑏𝑖𝑐𝑜𝑛𝑛}_𝑖𝛾^{𝑛−𝑝+1,𝑘−𝑞}_{𝑐𝑜𝑛𝑛}. (12) Then, 𝛾^𝑛,𝑘_{𝑐𝑜𝑛𝑛}=∑_{𝐺∈𝑉𝑋}^𝑛,𝑘𝛼_𝐺𝐺, where 𝛼_𝐺∈ℚ and 𝑋∶=∪^𝑛_𝑝=2∪^𝑘_𝑞=0𝑋^𝑝,𝑞. Moreover, for any equivalence class 𝒞∈ℰ(𝑉_𝑋^𝑛,𝑘) the following holds: (i) there exists 𝐺∈𝒞 such that 𝛼_𝐺>0 (ii) ∑_𝐺∈𝒞𝛼_𝐺=1/|𝑎𝑢𝑡𝒞|.

Proof. The result follows from the linearity of the mappings 𝑟^𝐺_𝑖 and the fact that larger graphs whose biconnected components are all in 𝑋 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 𝑉 be a vector space over ℚ. Let 𝖲(𝑉) denote the symmetric algebra on 𝑉. Then, 𝖲(𝑉)=⨁^∞_𝑘=0𝖲^𝑘(𝑉), where 𝖲⁰(𝑉)∶=ℚ1, 𝖲¹(𝑉)=𝑉, and 𝖲^𝑘(𝑉) is generated by the free commutative product of 𝑘 elements of 𝑉. Also, let 𝖲(𝑉)^⊗𝑛 denote the 𝑛-fold tensor product of 𝖲(𝑉) with itself. Recall that the multiplication in 𝖲(𝑉)^⊗𝑛 is given by the componentwise product: ⋅∶𝖲(𝑉)^⊗𝑛×𝖲(𝑉)^⊗𝑛→𝖲(𝑉)^⊗𝑛;𝑠₁⊗⋯⊗𝑠_𝑛,𝑠^′₁⊗⋯⊗𝑠^′_𝑛↦𝑠₁𝑠^′₁⊗⋯⊗𝑠_𝑛𝑠^′_𝑛, (13) where 𝑠_𝑖,  𝑠^′_𝑗 denote monomials on the elements of 𝑉 for all 𝑖,𝑗=1,…,𝑛. We may now proceed to the correspondence between graphs on {1,…,𝑛} and some elements of 𝖲(𝑉)^⊗𝑛. First, for all 𝑖,𝑗=1,…,𝑛 with 𝑖≠𝑗, we define the following tensors in 𝖲(𝑉)^⊗𝑛. 𝑅_𝑖,𝑗∶=1^⊗𝑖−1⊗𝑣⊗1^{⊗𝑗−𝑖−1}⊗𝑣⊗1^{⊗𝑛−𝑗}, (14) where 𝑣 is any vector different from zero. (As in Section 3, for simplicity, our notation does not distinguish between elements, say, 𝑅_𝑖,𝑗∈𝖲(𝑉)^⊗𝑛 and 𝑅_𝑖,𝑗∈𝖲(𝑉)^⊗𝑛′ with 𝑛≠𝑛^′. 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 𝑖,𝑗=1,…,𝑛 with 𝑖≠𝑗, let (i)a tensor factor in the 𝑖th position correspond to the vertex 𝑖 of a graph on {1,…,𝑛},(ii)a tensor 𝑅_𝑖,𝑗∈𝖲(𝑉)^⊗𝑛 correspond to the edge {𝑖,𝑗} of a graph on {1,…,𝑛}.

In this context, given a graph 𝐺 with 𝑉(𝐺)={1,…,𝑛} and 𝐸(𝐺)={{𝑖_𝑘,𝑗_𝑘}}_{𝑘=1,…,𝑚}, we define the following algebraic representation of graphs.(a)If 𝑚=0, then 𝐺 is represented by the tensor 1⊗⋯⊗1∈𝖲(𝑉)^⊗𝑛. (b)If 𝑚>0, then in 𝖲(𝑉)^⊗𝑛 the graph 𝐺 yields a monomial on the tensors which represent the edges of 𝐺. More precisely, since for all 1≤𝑘≤𝑚 each tensor 𝑅_{𝑖𝑘,𝑗𝑘}∈𝖲(𝑉)^⊗𝑛 represents an edge of the graph 𝐺, this is uniquely represented by the following tensor 𝑆^𝐺_1,…,𝑛∈𝖲(𝑉)^⊗𝑛 given by the componentwise product of the tensors 𝑅_{𝑖𝑘,𝑗𝑘}: 𝑆^𝐺_1,…,𝑛∶=^𝑚_𝑘=1𝑅_{𝑖𝑘,𝑗𝑘}. (15)Figure 4 shows some examples of this correspondence. Furthermore, let 𝑋⊆{1,…,𝑛^′} be a set of cardinality 𝑛 with 𝑛^′≥𝑛. Also, let 𝜎∶{1,…,𝑛}→𝑋 be a bijection. With this notation, define the following elements of 𝖲(𝑉)^⊗𝑛′: 𝑆^𝐺_{𝜎(1),…,𝜎(𝑛)}∶=^𝑚_𝑘=1𝑅_{𝜎(𝑖𝑘),𝜎(𝑗𝑘)}. (16) In terms of graphs, the tensor 𝑆^𝐺_{𝜎(1),…,𝜎(𝑛)} represents a disconnected graph, say, 𝐺^′, on the set {1,…,𝑛^′} consisting of a graph isomorphic to 𝐺 whose vertex set is 𝑋 and 𝑛^′−𝑛 isolated vertices in {1,…,𝑛^′}⧵𝑋. Figure 5 shows an example.

(a)

(b)

(c)

Figure 4 

(a) An isolated vertex represented by 1∈𝖲(𝑉). (b) A 2-vertex tree represented by 𝑅1,2∈𝖲(𝑉)⊗2. (c) A triangle represented by 𝑅1,2⋅𝑅1,3⋅𝑅2,3∈𝖲(𝑉)⊗3.

(a)

(b)

Figure 5 

(a) The graph represented by the tensor 𝑆𝐺1,2,3,4∈𝖲(𝑉)⊗4. (b) The graph represented by the tensor 𝑆𝐺1,5,4,2∈𝖲(𝑉)⊗5 associated with 𝐺 and the bijection 𝜎∶{1,2,3,4}→{1,2,4,5}; 1↦1,2↦5,3↦4,4↦2.

Furthermore, let 𝖳(𝖲(𝑉)) denote the tensor algebra on the graded vector space 𝖲(𝑉): 𝖳(𝖲(𝑉))∶=⨁^∞_𝑘=0𝖲(𝑉)^⊗𝑘. In 𝖳(𝖲(𝑉)) the multiplication •∶𝖳(𝖲(𝑉))×𝖳(𝖲(𝑉))→𝖳(𝖲(𝑉)) (17) is given by concatenation of tensors (e.g., see [20–22]): 𝑠₁⊗⋯⊗𝑠_𝑛•𝑠^′₁⊗⋯⊗𝑠^′_𝑛′∶=𝑠₁⊗⋯⊗𝑠_𝑛⊗𝑠^′₁⊗⋯⊗𝑠^′_𝑛′, (18) where 𝑠_𝑖,𝑠^′_𝑗 denote monomials on the elements of 𝑉 for all 𝑖=1,…,𝑛 and 𝑗=1,…,𝑛^′. We proceed to generalize the definition of the multiplication • to any two positions of the tensor factors. Let 𝜏∶S(𝑉)^⊗2→S(𝑉)^⊗2;𝑠₁⊗𝑠₂↦𝑠₂⊗𝑠₁. Moreover, define 𝜏_𝑘∶=1^⊗𝑘−1⊗𝜏⊗1^{⊗𝑛−𝑘−1}∶S(𝑉)^⊗𝑛→S(𝑉)^⊗𝑛 for all 1≤𝑘≤𝑛−1. In this context, for all 1≤𝑖≤𝑛, 1≤𝑗≤𝑛^′, we define •_𝑖,𝑗∶S(𝑉)^⊗𝑛×S(𝑉)^⊗𝑛′→S(𝑉)^{⊗𝑛+𝑛′} by the following equation: 𝑠₁⊗⋯⊗𝑠_𝑛•_𝑖,𝑗𝑠^′₁⊗⋯⊗𝑠^′_𝑛′∶=𝜏_𝑛−1∘⋯∘𝜏_𝑖𝑠₁⊗⋯⊗𝑠_𝑛⊗𝜏₁∘⋯∘𝜏_𝑗−1𝑠^′₁⊗⋯⊗𝑠^′_𝑛′=𝑠₁⊗⋯⊗𝑠_𝑖̂⊗⋯⊗𝑠_𝑛⊗𝑠_𝑖⊗𝑠^′_𝑗⊗𝑠^′₁⊗⋯⊗𝑠^′_𝑗⊗⋯⊗𝑠^′_𝑛′, (19) where 𝑠_𝑖̂ (resp. 𝑠^′_𝑗) means that 𝑠_𝑖 (resp. 𝑠^′_𝑗) is excluded from the sequence. In terms of the tensors 𝑆^𝐺_1,…,𝑛∈𝖲(𝑉)^⊗𝑛 and 𝑆^𝐺′_{1,…,𝑛′}∈𝖲(𝑉)^⊗𝑛′ the equation previous yields 𝑆^𝐺_1,…,𝑛•_𝑖,𝑗𝑆^𝐺′_{1,…,𝑛′}∶=𝑆^𝐺_{𝜎(1),…,𝜎(𝑛)}⋅𝑆^𝐺′_{𝜎′(1),…,𝜎′(𝑛′)}, (20) where 𝑆^𝐺_{𝜎(1),…,𝜎(𝑛)},𝑆^𝐺′_{𝜎′(1),…,𝜎′(𝑛′)}∈𝖲(𝑉)^{⊗𝑛+𝑛′}, 𝜎(𝑘)=𝑘 if 1≤𝑘<𝑖, 𝜎(𝑖)=𝑛, 𝜎(𝑘)=𝑘−1 if 𝑖<𝑘≤𝑛, and 𝜎^′(𝑘)=𝑘+𝑛+1 if 1≤𝑘<𝑗, 𝜎^′(𝑗)=𝑛+1, 𝜎^′(𝑘)=𝑘+𝑛 if 𝑗<𝑘≤𝑛^′. Clearly, the tensor 𝑆^𝐺_1,…,𝑛•_𝑖,𝑗𝑆^𝐺′_{1,…,𝑛′} represents a disconnected graph. Now, let ⋅_𝑖∶=1^⊗𝑖−1⊗⋅⊗1^{⊗𝑛−𝑖−1}∶𝖲(𝑉)^⊗𝑛→𝖲(𝑉)^⊗𝑛−1. In 𝖳(𝖲(𝑉)), for all 1≤𝑖≤𝑛, 1≤𝑗≤𝑛^′, define the following nonassociative and noncommutative multiplication: ⋄_𝑖,𝑗∶=⋅_𝑛∘•_𝑖,𝑗∶𝖲(𝑉)^⊗𝑛×𝖲(𝑉)^⊗𝑛′→𝖲(𝑉)^{⊗𝑛+𝑛′−1}. (21) The tensor 𝑆^𝐺_1,…,𝑛⋄_𝑖,𝑗𝑆^𝐺′_{1,…,𝑛′} represents the graph on 𝑛+𝑛^′−1 vertices, say, 𝐻 obtained by gluing the vertex 𝑖 of the graph 𝐺 to the vertex 𝑗 of the graph 𝐺^′. If both 𝐺 and 𝐺^′ are connected, the vertex 𝑛 is clearly a cutvertex of the graph 𝐻. Figure 6 shows an example.

(a)

(b)

Figure 6 

(a) The graphs represented by the tensors 𝑆𝐺1,2,3,4∈𝖲(𝑉)⊗4 and 𝑆𝐺′1,2,3∈𝖲(𝑉)⊗3. (b) The graph represented by the tensor 𝑆𝐻1,…,6=𝑆𝐺1,2,3,4⋄3,2𝑆𝐺′1,2,3∈𝖲(𝑉)⊗6.

6. Linear Mappings

We recall some of the linear mappings given in [9].

Let ℬ_𝑛,𝑘⊂𝖲(𝑉)^⊗𝑛 denote the vector space of all the tensors representing biconnected graphs in 𝑉^𝑛,𝑘_biconn. Let ℬ∶=ℬ_1,0⨁ℬ_2,0⨁^∞_𝑛=3⨁^{𝑘max(𝑛)}_𝑘=1ℬ_𝑛,𝑘⊂𝖳(𝖲(𝑉)), where 𝑘_max(𝑛)=(𝑛(𝑛−3)/2)+1 is, by Kirchhoff’s lemma [18], the maximum cyclomatic number of a biconnected graph on 𝑛 vertices. Let the mapping Δ∶ℬ→𝖳(𝖲(𝑉)) be given by the following equations: Δ(1)Δ𝐵∶=1⊗1,_{𝐺1,…,𝑛}1∶=𝑛^𝑛_𝑖=1Δ_𝑖𝐵^𝐺_1,…,𝑛if𝑛>1, (22) where 𝐺 denotes a biconnected graph on 𝑛 vertices represented by 𝐵^𝐺_1,…,𝑛∈𝖲(𝑉)^⊗𝑛. To define the mappings Δ_𝑖, we introduce the following bijections: (i)𝜎_𝑖()𝜈∶𝑗↦𝑗if1≤𝑗≤𝑖,𝑗+1if𝑖+1≤𝑗≤𝑛,ii_𝑖∶𝑗↦𝑗if1≤𝑗≤𝑖−1,𝑗+1if𝑖≤𝑗≤𝑛. (23) In this context, for all 𝑖=1,…,𝑛 with 𝑛>1, the mappings Δ_𝑖∶𝖲(𝑉)^⊗𝑛→𝖲(𝑉)^⊗𝑛+1 are defined by the following equation: Δ_𝑖𝐵^𝐺_1,…,𝑛∶=𝐵^𝐺_{𝜎𝑖(1),…,𝜎𝑖(𝑛)}+𝐵^𝐺_{𝜈𝑖(1),…,𝜈𝑖(𝑛)}=𝐵^𝐺_1,…,𝑖+1_{,𝑖+2,…,𝑛+1}+𝐵^𝐺_1,…,̂𝑖_{,𝑖+1…,𝑛+1}, (24) where 𝑖̂ (resp., 𝑖+1) means that the index 𝑖 (resp., 𝑖+1) is excluded from the sequence. The tensor 𝐵^𝐺_1,…,̂𝑖_,…,𝑛+1 (resp., 𝐵^𝐺_1,…,𝑖+1_{,𝑖+2,…,𝑛+1}) is constructed from 𝐵^𝐺_1,…,𝑛 by transferring the monomial on the elements of 𝑉 which occupies the 𝑘th tensor factor to the (𝑘+1)th position for all 𝑖≤𝑘≤𝑛 (resp., 𝑖+1≤𝑘≤𝑛). Furthermore, suppose that 𝐵^𝐺_{𝜋(1),…,𝜋(𝑛)}∈𝖲(𝑉)^⊗𝑛′ and that the bijection 𝜋 is such that 𝑖∉𝜋({1,…,𝑛})⊂{1,…,𝑛^′}. In this context, define Δ_𝑖𝐵^𝐺_{𝜋(1),…,𝜋(𝑛)}∶=𝐵^𝐺_{𝜈𝑖(𝜋(1)),…,𝜈𝑖(𝜋(𝑛))} (25) in agreement with Δ(1)∶=1⊗1. It is straightforward to verify that the mappings Δ_𝑖 satisfy the following property: Δ_𝑖∘Δ_𝑖=Δ_𝑖+1∘Δ_𝑖, (26) where we used the same notation for Δ_𝑖∶𝖲(𝑉)^⊗𝑛→𝖲(𝑉)^⊗𝑛+1 on the right of either side of the previous equation and Δ_𝑖∶𝖲(𝑉)^⊗𝑛+1→𝖲(𝑉)^⊗𝑛+2 as the leftmost operator on the left hand side of the equation. Accordingly, Δ_𝑖+1∶𝖲(𝑉)^⊗𝑛+1→𝖲(𝑉)^⊗𝑛+2 as the leftmost operator on the right hand side of the equation.

Now, for all 𝑚>0, define the 𝑚th iterate of Δ_𝑖, Δ^𝑚_𝑖∶𝖲(𝑉)^⊗𝑛→𝖲(𝑉)^⊗𝑛+𝑚, recursively as follows: Δ¹_𝑖∶=Δ_𝑖,Δ^𝑚_𝑖∶=Δ_𝑖∘Δ_𝑖^𝑚−1, (27)(28) where Δ_𝑖∶𝖲(𝑉)^{⊗𝑛+𝑚−1}→𝖲(𝑉)^⊗𝑛+𝑚 in formula (28). This can be written in 𝑚 different ways corresponding to the composition of Δ_𝑖^𝑚−1 with each of the mappings Δ_𝑗∶𝖲(𝑉)^{⊗𝑛+𝑚−1}→𝖲(𝑉)^⊗𝑛+𝑚 with 𝑖≤𝑗≤𝑖+𝑚−1. These are all equivalent by formula (26).

Extension to Connected Graphs. We now extend the mappings Δ_𝑖 to the vector space of all the tensors representing connected graphs ℬ^∗∶=⨁^∞_𝜇=0ℬ^⋄𝜇, where ℬ^⋄0=ℚ1. We proceed to define ℬ^⋄𝜇 for 𝜇>0.

First, given two sets 𝐴_𝑛⊂𝖲(𝑉)^⊗𝑛 and 𝐵_𝑝⊂𝖲(𝑉)^⊗𝑝, by 𝐴_𝑛⋄_𝑖,𝑗𝐵_𝑝⊂𝖲(𝑉)^{⊗𝑛+𝑝−1} with 𝑖=1,…,𝑛,𝑗=1,…,𝑝 we denote the set of elements obtained by applying the mapping ⋄_𝑖,𝑗 to every ordered pair (𝑎∈𝐴_𝑛,𝑏∈𝐵_𝑝). Also, let ℬ₂∶=ℬ_2,0 and ℬ_𝑛∶=⨁^{𝑘max(𝑛)}_𝑘=1ℬ_𝑛,𝑘, 𝑛≥3. In this context, for all 𝜇≥1 and 𝑛≥𝜇+1, define ℬ^∗_𝑛^𝜇 as follows: ℬ^∗_𝑛^𝜇=_{𝑛1+⋯+𝑛𝜇}^𝑛_{=𝑛+𝜇−1}¹_𝑖1=1⋯^𝑛𝜇−1_𝑖𝜇−1^𝑛₌₁^{𝜇+⋯+𝑛2−𝜇+2}_𝑗1=1⋯^𝑛𝜇_{𝑗𝜇−1=1}ℬ_𝑛1⋄_𝑖1,𝑗1ℬ_𝑛2⋄_𝑖2,𝑗2⋯ℬ_𝑛𝜇−2⋄_{𝑖𝜇−2,𝑗𝜇−2}ℬ_𝑛𝜇−1⋄_{𝑖𝜇−1,𝑗𝜇−1}ℬ_𝑛𝜇⋯. (29) Also, define ℬ^′_𝑛^∗𝜇=_{𝜋∈𝑆𝑛}𝑆^𝐺_{𝜋(1)(𝑛),…,𝜋}∣𝑆^𝐺_1,…,𝑛∈ℬ^∗_𝑛^𝜇, (30) where 𝑆_𝑛 denotes the symmetric group on the set {1,…,𝑛} and 𝑆^𝐺_{𝜋(1),…,𝜋(𝑛)} is given by formula (16). Finally, for all 𝜇≥1, define ℬ^⋄𝜇∶=^∞_𝑛=𝜇+1ℬ^′_𝑛^∗𝜇. (31) The elements of ℬ^⋄𝜇 are clearly tensors representing connected graphs on 𝜇 biconnected components. By (16) and (21), these may be seen as monomials on tensors representing biconnected graphs with the componentwise product · :𝖲(𝑉)^⊗𝑛×𝖲(𝑉)^⊗𝑛→𝖲(𝑉)^⊗𝑛 so that repeated indices correspond to cutvertices of the associated graphs. In this context, an arbitrary connected graph, say, 𝐺, on 𝑛≥2 vertices and 𝜇≥1 biconnected components yields ^𝜇_𝑎=1𝐵^𝐺𝑎_{𝜎𝑎(1),…,𝜎𝑎(𝑛𝑎)}, (32) Where, for all 1≤𝑎≤𝜇, 𝐵^𝐺𝑎_{𝜎𝑎(1),…,𝜎𝑎(𝑛𝑎)}∈𝖲(𝑉)^⊗𝑛, 𝐺_𝑎 is a biconnected graph on 2≤𝑛_𝑎≤𝑛 vertices represented by 𝐵^𝐺𝑎_{1,…,𝑛𝑎}∈𝖲(𝑉)^⊗𝑛𝑎, and 𝜎_𝑎∶{1,…,𝑛_𝑎}→𝑋_𝑎⊆{1,…,𝑛} is a bijection.

We now extend the mapping Δ∶=(1/𝑛)∑^𝑛_𝑖=1Δ_𝑖 to ℬ^∗ by requiring the mappings Δ_𝑖 to satisfy the following condition: Δ_𝑖^𝜇_𝑎=1𝐵^𝐺𝑎_{𝜎𝑎(1),…,𝜎𝑎𝑛𝑎}∶=^𝜇_𝑎=1Δ_𝑖𝐵^𝐺𝑎_{𝜎𝑎(1),…,𝜎𝑎𝑛𝑎}. (33) Given a connected graph 𝐺^′, the mapping Δ_𝑖 may be thought of as a way of (a) splitting the vertex 𝑖 into two new vertices numbered 𝑖 and 𝑖+1 and (b) distributing the biconnected components sharing the vertex 𝑖 between the two new ones in all the possible ways. Analogously, the action of the mappings Δ^𝑚_𝑖 consists of (a) splitting the vertex 𝑖 into 𝑚+1 new vertices numbered 𝑖,𝑖+1,…,𝑖+𝑚 and (b) distributing the biconnected components sharing the vertex 𝑖 between the 𝑚+1 new ones in all the possible ways.

We now combine the mappings Δ_𝑖^𝑛−1 with tensors representing biconnected graphs. Let 𝑛>1, 𝑝≥1 and 1≤𝑖≤𝑝 be fixed integers. Let 𝜋_𝑖∶{1,…,𝑛}→{𝑖,𝑖+1,…,𝑖+𝑛−1};𝑗↦𝑗+𝑖−1 be a bijection. Let 𝐺 be a biconnected graph on 𝑛 vertices represented by the tensor 𝐵^𝐺_1,…,𝑛∈𝖲(𝑉)^⊗𝑛. The tensors 𝐵^𝐺_{𝑖,𝑖+1,…,𝑖+𝑛−1}∶=𝐵^𝐺_{𝜋𝑖(1),…,𝜋𝑖(𝑛)}∈𝖲(𝑉)^{⊗𝑛+𝑝−1} (see formula (16)) may be viewed as operators acting on 𝖲(𝑉)^{⊗𝑛+𝑝−1} by multiplication. In this context, consider the following mappings given by the composition of 𝐵^𝐺_{𝑖,𝑖+1,…,𝑖+𝑛−1} with Δ_𝑖^𝑛−1: 𝐵^𝐺_{𝑖,𝑖+1,…,𝑖+𝑛−1}∘Δ_𝑖^𝑛−1∶𝖲(𝑉)^⊗𝑝→𝖲(𝑉)^{⊗𝑛+𝑝−1}. (34) These are the analog of the mappings 𝑟^𝐺_𝑖 given in Section 3. In plain English, the mappings 𝐵^𝐺_{𝑖,𝑖+1,…,𝑖+𝑛−1}∘Δ_𝑖^𝑛−1 produce a connected graph with 𝑛+𝑝−1 vertices from one with 𝑝 vertices in the following way:(a)split the vertex 𝑖 into 𝑛 new vertices, namely, 𝑖, 𝑖+1,…, 𝑖+𝑛−1, (b)distribute the biconnected components containing the split vertex between the 𝑛 new ones in all the possible ways, (c)merge the 𝑛 new vertices into the graph 𝐺. When the graph 𝐺 is a 2-vertex tree, the mapping 𝑅_1,2∘Δ coincides with the application 𝐿=(𝜙⊗𝜙)Δ of [23] when 𝜙 acts on 𝖲(𝑉) by multiplication with a vector.

To illustrate the action of the mappings 𝐵^𝐺_{𝜋𝑖(1),…,𝜋𝑖(𝑛)}∘Δ_𝑖, consider the graph 𝐻 consisting of two triangles sharing a vertex. Let this be represented by 𝐵^𝐶3_1,2,3⋅𝐵^𝐶3_3,4,5∈𝖲(𝑉)^⊗5, where 𝐶₃ denotes a triangle represented by 𝐵^𝐶3_1,2,3=𝑅_1,2⋅𝑅_2,3⋅𝑅_1,3∈𝖲(𝑉)^⊗3. Let 𝑇₂ denote a 2-vertex tree represented by 𝐵^𝑇2_1,2=𝑅_1,2∈𝖲(𝑉)^⊗2. Applying the mapping 𝑅_3,4∘Δ₃ to 𝐻 yields 𝑅_3,4∘Δ₃𝐵^𝐶3_1,2,3⋅𝐵^𝐶3_3,4,5=𝑅_3,4⋅Δ₃𝐵^𝐶3_1,2,3⋅Δ₃𝐵^𝐶3_3,4,5=𝑅_3,4⋅𝐵^𝐶3_1,2,3+𝐵^𝐶3_1,2,4⋅𝐵^𝐶3_3,5,6+𝐵^𝐶3_4,5,6=𝑅_3,4⋅𝐵^𝐶3_1,2,3⋅𝐵^𝐶3_3,5,6+𝑅_3,4⋅𝐵^𝐶3_1,2,3⋅𝐵^𝐶3_4,5,6+𝑅_3,4⋅𝐵^𝐶3_1,2,4⋅𝐵^𝐶3_3,5,6+𝑅_3,4⋅𝐵^𝐶3_1,2,4⋅𝐵^𝐶3_4,5,6. (35) 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 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.

Figure 7 

The linear combination of graphs obtained by applying the mapping 𝑅3,4∘Δ3 to the cutvertex of a graph consisting of two triangles sharing a vertex.

7. 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 𝑛≥2, trees on 𝑛 vertices are produced by connecting a vertex of a tree on 𝑖 vertices to a vertex of a tree on 𝑛−𝑖 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 𝐺 denote a connected graph. Recall the notation introduced in Section 4; 𝒱_𝐺 denotes the set of biconnected components of 𝐺 and ℰ(𝒱_𝐺) denotes the set of equivalence classes of the graphs in 𝒱_𝐺. Given a set 𝑋⊂∪^𝑛_𝑝=2∪^𝑘_𝑞=0𝑉^𝑝,𝑞_biconn, let 𝑉_𝑋^𝑛,𝑘∶={𝐺∈𝑉^𝑛,𝑘_conn∶ℰ(𝒱_𝐺)⊆ℰ(𝑋)} with the convention 𝑉_𝑋^1,0∶={({1},∅)}. With this notation, in the algebraic setting, Corollary 6 reads as follows.

Theorem 7. For all 𝑝>1 and 𝑞≥0, suppose that Φ^𝑝,𝑞_1,…,𝑝∶=∑_{𝐺′∈𝑋𝑝,𝑞}𝜎_𝐺′𝐵^𝐺′_1,…,𝑝∈ℬ_𝑝,𝑞⊂𝖲(𝑉)^⊗𝑝 with 𝜎_𝐺′∈ℚ and 𝑋^𝑝,𝑞⊆𝑉^𝑝,𝑞_{𝑏𝑖𝑐𝑜𝑛𝑛}, is such that for any equivalence class 𝒜∈ℰ(𝑋^𝑝,𝑞) the following holds: (i) there exists 𝐺^′∈𝒜 such that 𝜎_𝐺′>0, (ii) ∑_{𝐺′∈𝒜}𝜎_𝐺′=1/|𝑎𝑢𝑡𝒜|. In this context, for all 𝑛≥1 and 𝑘≥0, define Ψ^𝑛,𝑘∈𝖲(𝑉)^⊗𝑛 by the following recursion relation: Ψ^1,0Ψ∶=1,_1,𝑘Ψ∶=0if𝑘>0,_𝑛,𝑘1∶=×𝑘+𝑛−1_{𝑘𝑛𝑞=0𝑝=2𝑛−𝑝+1𝑖=1}𝑞+𝑝−1Φ^𝑝,𝑞_{𝑖,𝑖+1,…,𝑖+𝑝−1}⋅Δ_𝑖^𝑝−1Ψ^{𝑛−𝑝+1,𝑘−𝑞}. (36)(37) Then, Ψ^𝑛,𝑘=∑_{𝐺∈𝑉𝑋}^𝑛,𝑘𝛼_𝐺𝑆^𝐺_1,…,𝑛, where 𝛼_𝐺∈ℚ and 𝑋∶=∪^𝑛_𝑝=2∪^𝑘_𝑞=0𝑋^𝑝,𝑞. Moreover, for any equivalence class 𝒞∈ℰ(𝑉_𝑋^𝑛,𝑘), the following holds: (i) there exists 𝐺∈𝒞 such that 𝛼_𝐺>0, (ii) ∑_𝐺∈𝒞𝛼_𝐺=1/|𝑎𝑢𝑡𝒞|.

Now, given a nonempty set 𝑌⊆𝑉^𝑝,𝑞_biconn, we use the following notation: 𝑉^𝜇_𝑌∶=𝐺∈𝑉^{(𝑝−1)𝜇+1,𝑞𝜇}_conn∶ℰ𝒱_𝐺⊆ℰ(𝑌),card𝒱_𝐺=𝜇, (38) with the convention 𝑉⁰_𝑌∶={({1},∅)}. When we only consider graphs whose biconnected components are all in 𝑌, the previous recurrence can be transformed into one on the number of biconnected components.

Theorem 8. Let 𝑝>1 and 𝑞≥0 be fixed integers. Let 𝑌⊆𝑉^𝑝,𝑞_{𝑏𝑖𝑐𝑜𝑛𝑛} be a non-empty set. Suppose that 𝜙^𝑝,𝑞_1,…,𝑝∶=∑_{𝐺′∈𝑌}𝜎_𝐺′𝐵^𝐺′_1,…,𝑝∈ℬ_𝑝,𝑞⊂𝖲(𝑉)^⊗𝑝 with 𝜎_𝐺′∈ℚ, is such that for any equivalence class 𝒜∈ℰ(𝑌) the following holds: (i) there exists 𝐺^′∈𝒜 such that 𝜎_𝐺′>0, (ii) ∑_{𝐺′∈𝒜}𝜎_𝐺′=1/|𝑎𝑢𝑡(𝒜)|. In this context, for all 𝜇≥0, define 𝜓^𝜇∈𝖲(𝑉)^{⊗(𝑝−1)𝜇+1} by the following recursion relation: 𝜓⁰𝜓∶=1,_𝜇1∶=𝜇^{(𝑝−1)(𝜇−1)+1}_𝑖=1𝜙^𝑝,𝑞_{𝑖,𝑖+1,…,𝑖+𝑝−1}⋅Δ_𝑖^𝑝−1𝜓^𝜇−1. (39) Then, 𝜓^𝜇=∑_𝐺∈𝑉^𝜇_𝑌𝛼_𝐺𝑆^𝐺_{1,…,(𝑝−1)𝜇+1} with 𝛼_𝐺∈ℚ. Moreover, for any equivalence class 𝒞∈ℰ(𝑉^𝜇_𝑌), the following holds: (i) there exists 𝐺∈𝒞 such that 𝛼_𝐺>0, (ii) ∑_𝐺∈𝒞𝛼_𝐺=1/|𝑎𝑢𝑡(𝒞)|.

Proof. In this case, Ψ^𝑛,𝑘=0 unless 𝑛=(𝑝−1)𝜇+1 and 𝑘=𝑞𝜇, where 𝜇≥0. Therefore, the recurrence of Theorem 7 can be easily converted into a recurrence on the number of biconnected components by setting 𝜓^𝜇∶=Ψ^{(𝑝−1)𝜇+1,𝑞𝜇}.

For 𝑝=2 and 𝑞=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.

Proposition 9. For all 𝜇>0, 𝜓^𝜇=1𝜇×_{𝑖1+⋯+𝑖𝑝=𝜇−1}^{(𝑝−1)𝑖1+1}_𝑎1=1^{(𝑝−1)(𝑖1+𝑖2)+2}_{𝑎2=𝑖𝑝−11+2}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+𝑝}_{𝑎𝑝=(𝑝−1)(𝑖1+⋯+𝑖𝑝−1)+𝑝}𝜙_𝑎^𝑝,𝑞_{1,…,𝑎𝑝}⋅𝜓^𝑖1⊗⋯⊗𝜓^𝑖𝑝, (40) where 𝑖_𝑗=0,…,𝜇−1 for all 𝑗=1,…,𝑝.

Proof. Equation (40) is proved by induction on the number of biconnected components 𝜇. This is easily verified for 𝜇=1: 𝜓¹=𝜙^𝑝,𝑞_1,…𝑝⋅(1⊗⋯⊗1)=𝜙^𝑝,𝑞_1,…𝑝. (41) We now assume the formula to hold for 𝜓^𝜇−1. Then, formula (37) yields 𝜓^𝜇=1𝜇^{(𝑝−1)(𝜇−1)+1}_𝑖=1𝜙^𝑝,𝑞_{𝑖,𝑖+1,…,𝑖+𝑝−1}⋅Δ_𝑖^𝑝−1𝜓^𝜇−1=1𝜇𝜇−1^{(𝑝−1)(𝜇−1)+1}_𝑖=1𝜙^𝑝,𝑞_{𝑖,𝑖+1,…,𝑖+𝑝−1}⋅Δ_𝑖^𝑝−1_{𝑖1+⋯+𝑖𝑝=𝜇−2(𝑝−1)𝑖1+1𝑎1=1(𝑝−1)(𝑖1+𝑖2)+2𝑎2=𝑖𝑝−11+2}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+𝑝}_{𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+𝑝}𝜙_𝑎^𝑝,𝑞_{1,…,𝑎𝑝}⋅𝜓^𝑖1⊗⋯⊗𝜓^𝑖𝑝=1𝜇𝜇−1×_{𝑖1+⋯+𝑖𝑝=𝜇−2}^{(𝑝−1)(𝜇−1)+1}_𝑖=1Δ_𝑖^{𝑝−1(𝑝−1)𝑖1+1}_𝑎1=1^{(𝑝−1)(𝑖1+𝑖2)+2}_{𝑎2=𝑖𝑝−11+2}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+𝑝}_{𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+𝑝}𝜙_𝑎^𝑝,𝑞_{1,…,𝑎𝑝}⋅𝜙^𝑝,𝑞_{𝑖,𝑖+1,…,𝑖+𝑝−1}⋅Δ_𝑖^𝑝−1𝜓^𝑖1⊗⋯⊗𝜓^𝑖𝑝=1𝜇×𝜇−1_{𝑖1+⋯+𝑖𝑝=𝜇−2(𝑝−1)𝑖1+1𝑖=1𝑖}^𝑝−1_{(𝑝−1)𝑖1+1𝑎}Δ_{1=1(𝑝−1)(𝑖1+𝑖2)+2𝑎2=𝑖𝑝−11+2}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+𝑝}_{𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+𝑝(𝑝−1)(𝑖1+⋯+𝑖𝑝)+𝑝𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+𝑝}𝜙_𝑎^𝑝,𝑞_{1,…,𝑎𝑝}⋅𝜙^𝑝,𝑞_{𝑖,𝑖+1,…,𝑖+𝑝−1}⋅Δ_𝑖^𝑝−1𝜓^𝑖1⊗⋯⊗𝜓^𝑖𝑝+^{(𝑝−1)(𝑖1+𝑖2)+2}_{𝑖𝑖=𝑝−11+2}Δ_𝑖^𝑝−1_{(𝑝−1)𝑖1+1𝑎1=1(𝑝−1)(𝑖1+𝑖2)+2𝑎2=𝑖𝑝−11+2}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+𝑝}_{𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+𝑝}⋅𝜙^𝑝,𝑞_{𝑖,𝑖+1,…,𝑖+𝑝−1}⋅𝜓^𝑖1⊗Δ_𝑖^𝑝−1𝜓^𝑖2⊗⋯⊗𝜓^𝑖𝑝+⋯+^{(𝑝−1)(𝑖+1+⋯+𝑖𝑝)+𝑝}_{𝑖𝑖=𝑝−11+𝑖2+⋯+𝑖𝑝−1+𝑝}Δ_𝑖^𝑝−1_{(𝑝−1)𝑖1+1𝑎1=1(𝑝−1)(𝑖1+𝑖2)+2𝑎2=𝑖𝑝−11+2}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+𝑝}_{𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+𝑝}𝜙_𝑎^𝑝,𝑞_{1,…,𝑎𝑝}⋅𝜙^𝑝,𝑞_{𝑖,𝑖+1,…,𝑖+𝑝−1}⋅𝜓^𝑖1⊗⋯⊗Δ_𝑖^𝑝−1𝜓^𝑖𝑝=1𝜇×𝜇−1_{𝑖1+⋯+𝑖𝑝=𝜇−2}𝑖₁×+1_{(𝑝−1)𝑖1+𝑝𝑎1=1(𝑝−1)(𝑖1+𝑖2)+𝑝+1𝑎2=𝑖𝑝−11+𝑝+1}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+2𝑝−1}_{𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+2𝑝−1}𝜙_𝑎^𝑝,𝑞_{1,…,𝑎𝑝}⋅𝜓^𝑖1+1⊗⋯⊗𝜓^𝑖𝑝+𝑖₂×+1_{(𝑝−1)𝑖1+1𝑎1=1(𝑝−1)(𝑖1+𝑖2)+𝑝+1𝑎2=𝑖𝑝−11+2}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+2𝑝−1}_{𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+2𝑝−1}𝜙_𝑎^𝑝,𝑞_{1,…,𝑎𝑝}⋅𝜙^𝑝,𝑞_{𝑖,𝑖+1,…,𝑖+𝑝−1}⋅𝜓^𝑖1⊗𝜓^𝑖2+1⊗⋯⊗𝜓^𝑖𝑝+⋯+𝑖_𝑝×+1_{(𝑝−1)𝑖1+1𝑎1=1(𝑝−1)(𝑖1+𝑖2)+2𝑎2=𝑖𝑝−11+2}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+𝑝}_{𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+2𝑝−1}𝜙_𝑎^𝑝,𝑞_{1,…,𝑎𝑝}⋅𝜙^𝑝,𝑞_{𝑖,𝑖+1,…,𝑖+𝑝−1}⋅𝜓^𝑖1⊗⋯⊗𝜓^𝑖𝑝+1=1𝜇×𝜇−1_{𝜇−1𝑖1=0𝑖2+⋯+𝑖𝑝=𝜇−1−𝑖1}𝑖₁+^𝜇−1_{𝑖2=0𝑖1+𝑖3+⋯+𝑖𝑝=𝜇−1−𝑖2}𝑖₂+⋯+^𝜇−1_{𝑖𝑝=0𝑖1+⋯+𝑖𝑝−1=𝜇−1−𝑖𝑝}𝑖_𝑝×^{(𝑝−1)𝑖1+1}_𝑎1=1^{(𝑝−1)(𝑖1+𝑖2)+2}_{𝑎2=𝑖𝑝−11+2}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+𝑝}_{𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+𝑝}𝜙_𝑎^𝑝,𝑞_{1,…,𝑎𝑝}⋅𝜓^𝑖1⊗⋯⊗𝜓^𝑖𝑝=1𝜇_{𝑖2+⋯+𝑖𝑝=𝜇−1}^{(𝑝−1)𝑖1+1}_𝑎1=1^{(𝑝−1)(𝑖1+𝑖2)+2}_{𝑎2=𝑖𝑝−11+2}⋯^{(𝑝−1)(𝑖1+⋯+𝑖𝑝)+𝑝}_{𝑎𝑝=𝑖𝑝−11+⋯+𝑖𝑝−1+𝑝}𝜙_𝑎^𝑝,𝑞_{1,…,𝑎𝑝}⋅𝜓^𝑖1⊗⋯⊗𝜓^𝑖𝑝. (42)

Note that formula (40) states that the linear combination of all the equivalence classes of connected graphs in 𝑉^𝜇_𝑌 is given by summing over all the 𝑝-tuples of connected graphs with total number of biconnected components equal to 𝜇−1, gluing a vertex of each of them to distinct vertices of a graph in ℰ(𝑌) in all the possible ways.

8. Cayley-Type Formulas

Let 𝑌⊆𝑉^𝑝,𝑞_biconn. Recall that 𝑉^𝜇_𝑌 is the set of connected graphs on 𝜇 biconnected components, each of which has 𝑝 vertices and cyclomatic number 𝑞 and is isomorphic to a graph in 𝑌. We proceed to use formula (40) to recover some enumerative results which are usually obtained via generating functions and the Lagrange inversion formula, see [3]. In particular, we extend to graphs in 𝑉^𝜇_𝑌 the result that the sum of the inverses of the orders of the groups of automorphisms of all the pairwise nonisomorphic trees on 𝑛 vertices equals 𝑛^𝑛−2/𝑛! [12, page 209].

Proposition 10. Let 𝑝>1 and 𝑞≥0 be fixed integers. Let 𝑌⊆𝑉^𝑝,𝑞_{𝑏𝑖𝑐𝑜𝑛𝑛} be a non-empty set. For all 𝜇≥0, _{𝑉𝒞∈ℰ𝜇𝑌}1||=𝑎𝑢𝑡𝒞𝑝−1𝜇+1^𝜇−2𝜇!𝑝_{(𝑌)𝒜∈ℰ}1|𝑎𝑢𝑡𝒜|^𝜇. (43)

Proof. We first recall the following well-known identities derived from Abel’s binomial theorem [14]: ^𝑛−1_𝑖=1𝑛𝑖𝑖^𝑖−1(𝑛−𝑖)^{𝑛−𝑖−1}=2(𝑛−1)𝑛^𝑛−2,_𝑛𝑖=0𝑛𝑖(𝑥+𝑖)^𝑖−1𝑦+(𝑛−𝑖)^{𝑛−𝑖−1}=1𝑥+1𝑦𝑥+𝑦+𝑛^𝑛−1, (44)(45) where 𝑥 and 𝑦 are non-zero numbers. A proof may be found in [15] for instance. From (45) follows that for all 𝑝>1 and non-zero numbers 𝑥₁,…,𝑥_𝑝, the identity _{𝑖1+⋯+𝑖𝑝=𝑛}𝑛𝑖₁,…,𝑖_𝑝𝑥₁+𝑖₁^𝑖1−1⋯𝑥_𝑝+𝑖_𝑝^𝑖𝑝−1=𝑥₁+⋯+𝑥_𝑝𝑥₁⋯𝑥_𝑝𝑥₁+⋯+𝑥_𝑝+𝑛^𝑛−1 (46) holds. The proof proceeds by induction. For 𝑝=2 the identity specializes to (45). We assume the identity (46) to hold for 𝑝−1. Then, _{𝑖1+⋯+𝑖𝑝=𝑛}𝑛𝑖₁,…,𝑖_𝑝𝑥₁+𝑖₁^𝑖1−1⋯𝑥_𝑝+𝑖_𝑝^𝑖𝑝−1=^𝑛_{𝑖1=0𝑖2+⋯+𝑖𝑝=𝑛−𝑖1}𝑛𝑖₁𝑛−𝑖₁𝑖₂,…,𝑖_𝑝×𝑥₁+𝑖₁^𝑖1−1⋯𝑥_𝑝+𝑖_𝑝^𝑖𝑝−1=𝑥₂+⋯+𝑥_𝑝𝑥₂⋯𝑥_𝑝×^𝑛_𝑖1=0𝑛𝑖₁𝑥₁+𝑖₁^𝑖1−1𝑥₂+⋯+𝑥_𝑝+𝑛−𝑖₁^{𝑛−𝑖1−1}=𝑥₂+⋯+𝑥_𝑝𝑥₂⋯𝑥_𝑝×1𝑥₁+1𝑥₂+⋯+𝑥_𝑝𝑥₁+⋯+𝑥_𝑝+𝑛^𝑛−1=𝑥₁+⋯+𝑥_𝑝𝑥₁⋯𝑥_𝑝𝑥₁+⋯+𝑥_𝑝+𝑛^𝑛−1. (47) We turn to the proof of formula (43). Let 𝐼(𝜇)∶=∑_{𝒞∈ℰ(𝑉}^𝜇_𝑌)(1/|aut𝒞|). Formula (40) induces the following recurrence for 𝐼(𝜇): 𝐼(0)𝐼𝜇=1=1,𝜇_{(𝑌)𝒜∈ℰ}1|aut𝒜|×_{𝑖1+⋯+𝑖𝑝=𝜇−1}𝑝−1𝑖₁⋯𝑖+1𝑝−1_𝑝+1𝐼𝑖₁⋯𝐼𝑖_𝑝. (48) We proceed to prove by induction that 𝐼(𝜇)=(((𝑝−1)𝜇+1)^𝜇−2/𝜇!)(𝑝∑_{𝒜∈ℰ(𝑌)}(1/|aut𝒜|))^𝜇. The result holds for 𝜇=0,1: 𝐼(0)𝐼(1)=1,=_{𝒜∈ℰ(𝑌)}1||.aut𝒜 (49) We assume the result to hold for all 0≤𝑖≤𝜇−1. Then, 𝐼𝜇=1𝜇_{𝑖1+⋯+𝑖𝑝=𝜇−1}𝑝−1𝑖₁⋯𝑖+1𝑝−1_𝑝+1𝐼𝑖₁⋯𝐼𝑖_{𝑝𝒜∈ℰ(𝑌)}1||=𝑝aut𝒜_𝜇−1𝜇_{𝑖1+⋯+𝑖𝑝=𝜇−1}1𝑖₁!⋯𝑖_𝑝!𝑝−1𝑖₁+1^𝑖1−1⋯𝑝−1𝑖_𝑝+1^𝑖𝑝−1_{(𝑌)𝒜∈ℰ}1|aut𝒜|^𝜇=𝑝^𝜇−1𝜇!𝑝−1^{𝜇−1−𝑝}×_{𝑖1+⋯+𝑖𝑝=𝜇−1}𝑖𝜇−1₁,…,𝑖_𝑝𝑖₁+1𝑝−1^𝑖1−1⋯𝑖_𝑝+1𝑝−1^𝑖𝑝−1_{(𝑌)𝒜∈ℰ}1|aut𝒜|^𝜇=1𝜇!𝑝−1^{𝜇−1−𝑝}𝑝−1^𝑝−1×𝑝𝜇−1+𝑝−1^𝜇−2𝑝_{(𝑌)𝒜∈ℰ}1|aut𝒜|^𝜇=1𝜇!𝑝−1𝜇+1^𝜇−2𝑝_{(𝑌)𝒜∈ℰ}1|aut𝒜|^𝜇, (50)(51)(52)(53)(54) where we used formula (46) in going from (52) to (53). This completes the proof of Proposition 10.

The corollary is now established.

Corollary 11. Let 𝑝>1 and 𝑞≥0 be fixed integers. Let 𝑌⊆𝑉^𝑝,𝑞_{𝑏𝑖𝑐𝑜𝑛𝑛} be a non-empty set. For all 𝜇≥0, {𝐺∈𝑉^𝜇_𝑌∣𝑉(𝐺)={1,…,(𝑝−1)𝜇+1} is a set of cardinality 𝑝−1𝜇+1!𝑝−1𝜇+1^𝜇−2𝜇!𝑝_{(𝑌)𝒜∈ℰ}1|𝑎𝑢𝑡𝒜|^𝜇. (55)

Proof. The result is a straightforward application of Lagrange’s theorem to the symmetric group on the set {1,…,(𝑝−1)𝜇+1} and each of its subgroups aut𝒞, forall𝒞∈ℰ(𝑉^𝜇_𝑌).

For 𝑝=2 and 𝑞=0, formula (55) specializes to Cayley’s formula [11]: 𝜇+1^𝜇−1=𝑛^𝑛−2, (56) where 𝑛=𝜇+1 is the number of vertices of all the trees with 𝜇 biconnected components. In this particular case, the recurrence given by formula (48) can be easily transformed into a recurrence on the number of vertices 𝑛: 𝐽(1)𝐽(𝑛)=1=1,2(𝑛−1)^𝑛−1_𝑖=1𝑖(𝑛−𝑖)𝐽(𝑖)𝐽(𝑛−𝑖), (57) which by (44) yields 𝐽(𝑛)=𝑛^𝑛−2/𝑛!. Now, for 𝑇(𝑛)=𝑛!𝐽(𝑛), we obtain Dziobek’s recurrence for Cayley’s formula [10, 24]: 𝑇(1)𝑇(𝑛)=1=1,2(𝑛−1)^𝑛−1_𝑖=1𝑛𝑖𝑖(𝑛−𝑖)𝑇(𝑖)𝑇(𝑛−𝑖). (58) Furthermore, for graphs whose biconnected components are all complete graphs on 𝑝 vertices, formula (55) yields 𝑝−1𝜇+1!𝑝−1𝜇+1^𝜇−2𝑝−1!^𝜇𝜇! (59) in agreement with Husimi’s result for this particular case [25] (see also [26]). Also, when the biconnected components are all cycles of length 𝑝, we recover a particular case of Leroux’s result [27]: 𝑝−1𝜇+1!𝑝−1𝜇+1^𝜇−22^𝜇.𝜇! (60)

Acknowledgment

The research was supported through the fellowship SFRH/BPD/48223/2008 provided by the Portuguese Foundation for Science and Technology (FCT).