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.

As pointed out in [

Furthermore, the problem of generating graphs taking into account their symmetries was considered as early as the 19th century [

For simplicity, here by

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 [

Furthermore, we prove that when we only consider the restricted class of connected graphs whose biconnected components all have, say,

This paper is organized as follows. Section

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

Here, a

Moreover, given a graph

We now introduce a definition of labeled graph. Let

Furthermore, an

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

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

where the graphs

The mappings

Furthermore, let

We proceed to generalize the edge contraction operation given in [

The linear combination of graphs obtained by applying the mapping

where the graph

For instance, Figure

We now introduce the following auxiliary mapping.

Let

where the graphs

The mapping

The graph obtained by applying the mapping

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

We proceed to generalize the recursion formula for generating trees given in [

For all

The proof is very analogous to the one given in [

Let

The proof proceeds by induction on the number of biconnected components

Let

The proof proceeds by induction on the number of biconnected components

Choose any one of the

We now show that

Let

Let

Let

Let

This completes the proof of Theorem

Figure

The result of computing all the pairwise non-isomorphic connected graphs as contributions to

For all

The result follows from the linearity of the mappings

We represent graphs by tensors whose indices correspond to the vertex numbers. Our description is essentially that of [

Let

a tensor factor in the

a tensor

In this context, given a graph

If

If

(a) An isolated vertex represented by

(a) The graph represented by the tensor

Furthermore, let

(a) The graphs represented by the tensors

We recall some of the linear mappings given in [

Let

Now, for all

First, given two sets

We now extend the mapping

We now combine the mappings

split the vertex

distribute the biconnected components containing the split vertex between the

merge the

To illustrate the action of the mappings

The linear combination of graphs obtained by applying the mapping

Reference [

Let

For all

Now, given a nonempty set

Let

In this case,

For

For all

Equation (

Note that formula (

Let

Let

We first recall the following well-known identities derived from Abel’s binomial theorem [

The corollary is now established.

Let

The result is a straightforward application of Lagrange’s theorem to the symmetric group on the set

For

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