2-regular Digraphs of the Lovelock Lagrangian

TL;DR

This paper catalogs all 2-regular directed graphs on up to 9 nodes, which correspond to terms in the Lovelock Lagrangian involving contractions of Riemann tensors, including various graph types.

Contribution

It provides a comprehensive enumeration of 2-regular digraphs relevant to the Lovelock Lagrangian, including disconnected, multiarc, and loop graphs, for the first time.

Findings

Tabulated arc lists for n=0 to 9 nodes

Includes graphs with loops and multiarcs

Links graphs to Lovelock Lagrangian terms

Abstract

The manuscripts tabulates arc lists of the 1, 1, 3, 8, 25, 85, 397 ... unlabeled 2-regular digraphs on n=0, 1, 2, ..., 9 nodes, including disconnected graphs, graphs with multiarcs and/or graphs with loops. Each of these graphs represents one term of the Lagrangian of Lovelock's type -- a contraction of a product of n Riemann tensors -- once the 2 covariant and 2 contravariant indices of a tensor are associated with the in-edges and out-edges of a node.

Tables

Table 1. Table 1: Labeled 2-regular digraphs L 2 ​ ( n , c ) subscript 𝐿 2 𝑛 𝑐 L_{2}(n,c) with n 𝑛 n nodes and c 𝑐 c (weak) components, allowing loops and multiarcs (Foundation Inc., 2023 , A307804) .

$n\backslash c$	$1$	$2$	3	4	5	6	7	$L_2\left(n\right)$
1	1							1
2	2	1						3
3	14	6	1					21
4	201	68	12	1				282
5	4704	1285	200	20	1			6210
6	160890	36214	4815	460	30	1		202410
7	7538040	1422288	160594	13755	910	42	1	9135630

Table 2. Table 2: Connected unlabeled 2-regular digraphs after cutting one arc, and corresponding labeled 2-regular digraphs (one or any number of components) allowing loops and multiarcs.

$n$	$U_{2,+2}(n,1)$	$L_{2,+2}(n,1)$	$L_{2,+2}\left(n\right)$
2	1	2	2
3	1	6	12
4	3	72	132
5	12	1440	2400
6	61	43200	66060
7	365	1806840	2572920

Table 3. Table 3: Connected unlabeled 2-regular digraphs after cutting two arcs, and corresponding labeled 2-regular digraphs (one or any number of components) allowing loops and multiarcs.

$n$	$U_{2,+4}(n,1)$	$L_{2,+4}(n,1)$	$L_{2,+4}\left(n\right)$
4	0	0	0
5	1	30	210
6	4	1260	5220
7	21	60480	180810

Table 4. Table 4: Unlabeled 2-regular digraphs U 2 ​ ( n , c ) subscript 𝑈 2 𝑛 𝑐 U_{2}(n,c) with n 𝑛 n nodes and c 𝑐 c (weak) components, allowing loops and multiarcs (Foundation Inc., 2023 , A306892,A006372) .

$n\backslash c$	$1$	$2$	3	4	5	6	7	8	9	$U_2\left(n\right)$
1	1									1
2	2	1								3
3	5	2	1							8
4	14	8	2	1						25
5	50	24	8	2	1					85
6	265	93	28	8	2	1				397
7	1601	435	108	28	8	2	1			2183
8	11984	2486	507	113	28	8	2	1		15129
9	101884	17211	2811	527	113	28	8	2	1	122585

Table 5. Table 5: Unlabeled 2-regular digraphs U 2 ( r ) ​ ( n ) = U 2 ( n − r ) ​ ( n ) superscript subscript 𝑈 2 𝑟 𝑛 superscript subscript 𝑈 2 𝑛 𝑟 𝑛 U_{2}^{(r)}(n)=U_{2}^{(n-r)}(n) with n 𝑛 n nodes and 0 ≤ r ≤ n 0 𝑟 𝑛 0\leq r\leq n rooted nodes, allowing loops and multiarcs. U 2 ( 0 ) ​ ( n ) = U 2 ​ ( n ) superscript subscript 𝑈 2 0 𝑛 subscript 𝑈 2 𝑛 U_{2}^{(0)}(n)=U_{2}(n) .

$n\backslash r$	$0$	$1$	$2$	3	4	5	6
1	1	1
2	3	3	3
3	8	13	13	8
4	25	58	88	58	25
5	85	310	588	588	310	85
6	397	1909	4626	6035	4626	1909	397
7	2183	13843	40417	66471	66471	40417	13843
8	15129	114821	395324	782257	975715	782257	395324

Equations

$$U_k(n,c)=\sum _{\stackrel{n=n_1+2n_2+\cdots +c{n}_c}{n_i\ge 0}}\prod _{i=1}^c\left(\genfrac{}{}{0ex}{}{U_k(i,1)+n_i-1}{n_i}\right).$$

$$U_k(0)=1;U_k(n)$$

$$L_k(0)=1;L_k(n)$$

$$U_1(n,1)=1.$$

$$L_k(n,c)=\frac{1}{c!}\sum _{\stackrel{n=n_1+n_2+\cdots +n_c}{n_i\ge 1}}\left(\genfrac{}{}{0ex}{}{n}{n_1,n_2,\cdots ,n_c}\right)\prod _{i=1}^c{L}_k(n_i,1).$$

$$L_k(x,1)\equiv \sum _{n\ge 1}\frac{L_k(n,1)}{n!}{x}^n$$

$$L_k(x,t)\equiv \sum _{n,c\ge 0}\frac{L_k(n,c)}{n!}{x}^n{t}^c=\exp \left[t{L}_k(x,1)\right].$$

Full text

digraphs, regular graphs, Lovelock Lagrangian

I Regular Digraphs

I.1 Nomenclature

In the common language of graph theory, digraphs (directed graphs) are graphs where the edges (called arcs) are oriented, i.e., the two nodes of an arc are distinguished to be tail and head of the arc. The indegree of a node is the number of arcs that have heads at a node; the outdegree is the number of arcs that have tails at a node. $k$-regular digraphs are digraphs where the indegree and outdegree at each node is the same $k$ Brinkmann (2013); Ramanath and Walsh (1987). Loops are arcs where head and tail are the same node. (Each loop increases the indegree and outdegree of the node by one.) Multiarcs are multisets of two or more arcs that share a head and tail node.

The underlying simple graph of a digraph is obtained by reducing the arcs to undirected edges, replacing multiedges by single edges, and removing loops. We will be only interested in these simple graphs to classify the digraphs by the number of components of their underlying simple graphs. (That means according to weak connectivity).

(Vertex) labeled graphs have distinct labels at their nodes, usually taken to be the positive integers while handling graphs on computers, or letters from a onwards. We define the Adjacency Matrix of a digraph of nodes labeled $1,2,\ldots$ as the square matrix which contains in row $r$ and column $c$ the number of arcs which start at node $r$ and end at node $c$. Construction of the labeled $k$-regular digraphs admitting multiarcs and loops is therefore equivalent to constructing the $n\times n$ matrices with non-negative integer entries where all row sums and all column sums are $k$. These are registered in Table A008300 in the Online Encyclopedia of Integer Sequences for the cases where multiarcs are not admitted, and in Table A257493 if multiarcs are admitted Foundation Inc. (2023).

Denote the number of unlabeled and labeled $k$-regular digraphs with $n$ nodes and $c$ components by $U_{k}(n,c)$ and $L_{k}(n,c)$.

The unlabeled graphs with $c>1$ components can be derived from the connected graphs quickly by the Multiset Transformation, collecting arrangements over all partitions of $n$ into $c$ parts (Flajolet and Sedgewick, 2009, Theor. I.1)(Harary and Palmer, 1973, (4.2.3)):

[TABLE]

The number of unlabeled and labeled $k$-regular digraphs with $n$ nodes is

[TABLE]

The case of the unlabeled 1-regular digraphs is simple: the connected 1-regular digraphs are cycles (unicycles), including the case with 1 node and its loop:

[TABLE]

So the number of unlabeled, not necessarily connected, 1-regular digraphs on $n$ nodes, $U_{1}(n)$, is the number of partitions of $n$ (Foundation Inc., 2023, A000041), and the $U_{1}(n,c)$ are the partition numbers (Foundation Inc., 2023, A008284).

I.2 Symmetries

The key part of this work is to identify the Automorphism Group of the labeled 2-regular digraphs, which is the group of permutations of the labels which keeps the Adjacency Matrix of a graph the same. This bundles a set of one or more labeled digraphs which are mapped onto each other by the permutations of the group, and each such set is represented by a single unlabeled 2-regular digraph. This is one way of erasing/forgetting any particular order on the nodes while maintaining the connectivity and structure of the graph.

The unlabeled 2-regular digraphs might also be obtained by starting from the cubic QED vacuum polarization diagrams de Mello Koch and Ramgoolam (2012); Mathar (2019), coalescing each edge that represents an undirected photon line and its two incident nodes into a single node such that only the directed fermion lines connect to the remaining nodes—representing nodes of a $\varphi^{4}$ theory—, plus a cleanup that eliminates duplicates. Another approach is to construct first the 4-regular undirected unlabeled graphs de Mello Koch and Ramgoolam (2012)(Foundation Inc., 2023, A129429), and to add directions later.

We shall keep track of the Automorphism Group $\cal A$ of each graph by writing down the Cycle Index of the group of the node permutations (Harary and Palmer, 1973, §2). Since $\cal A$ is a subgroup of the group of all permutations of $n$, its order, $|\cal A|$, is a divisor of $n!$. The denominator of the cycle index polynomial is $|\cal A|$, so we can recover the number of labeled digraphs by dividing $n!$ through the denominator of the Cycle Index (Harary and Palmer, 1973, (1.1.3)). For $n=3$ nodes, for example, $U_{2}(3)=8$ unlabeled 2-regular digraphs exist with 4 different Cycle Indices:

• •

1 graph with cycle index $(t_{1}^{3})/1$,

• •

3 graphs with cycle index $(t_{1}^{3}+t_{1}t_{2})/2$,

• •

2 graphs with cycle index $(t_{1}^{3}+2t_{3})/3$,

• •

and 2 graphs with cycle index $(t_{1}^{3}+3t_{1}t_{2}+2t_{3})/6$,

and $1\times 3!/1+3\times 3!/2+2\times 3!/3+2\times 3!/6=21=L_{2}(3)$ is the number of labeled graphs on 3 nodes (Foundation Inc., 2023, A000681). As we are admitting loops, there is always at least one graph on $n$ nodes (the one consisting of $n$ isolated nodes with two loops each) that has the maximum symmetry here, $|{\cal A}|=n!$.

The labeled graphs with $c>1$ components are deduced from the weakly connected labeled graphs by a Bell transformation, summing over all compositions of $n$ into positive parts weighted by multinomial coefficients (Bower, , EIJ):

[TABLE]

If

[TABLE]

denotes the exponential generating function of the weakly connected labeled graphs, the bivariate exponential generating function of the labeled graphs is Gilbert (1956)

[TABLE]

The corresponding derivation starting from $L_{1}(n,1)=1$ demonstrates that $L_{1}(n,c)$ are the Stirling Numbers of the Second Kind (Foundation Inc., 2023, A008277) and $L_{1}(n)$ the Bell Numbers (Foundation Inc., 2023, A000110).

For $L_{2}$, the number of labeled digraphs refined by the number of weakly connected components is summarized in Table 1.

II Gallery of Unlabeled 2-regular Digraphs

The unlabeled 2-regular digraphs on $n\leq 5$ nodes are illustrated in the following sections. Each digraph has $2n$ arcs. For easier visual recognition, the graphs with more than one component are surrounded by a frame. These visualize throw schemes for $n$ 2-armed jugglers.

II.1 1 graph on 1 node

II.2 3 graphs (2 connected) on 2 nodes

II.3 8 graphs (5 connected) on 3 nodes

II.4 25 graphs (14 connected) on 4 nodes

II.5 85 graphs (50 connected) on 5 nodes

III Bijection with Lovelock Terms

Bogdanos has pointed at a graphical representation of the terms of Lovelock’s Lagrange density Lovelock (1971, 1972); Bogdanos (2009): A graph of a particular term which is a product of $n$ Riemann Tensors is constructed as follows: (i) Start with a bottom row of $n$ nodes plus a top row of another $n$ nodes, where nodes are associated left-to-right with a left-to-right reading of the $R$-factors of the product. (ii) Add an edge from the $b$-th bottom node to the $t$-th top node if a covariant lower index of factor number $b$ appears as a contravariant index of factor number $t$. This assignment works because the two contravariant indices are permutations of the covariant indices. Since the Riemann Tensor is (skew)-symmetric in the first and in the last two indices, the representation does not need to track the two edges at each node to distinguish a “first” from a “second.” Bogdanos’ representation is a labeled, undirected 2-regular graph on $2n$ nodes.

Our graphical representation transforms these bipartite graphs once more Cooper (1993):

• •

Bogdanos’ edges are turned into directed edges (arcs), always heading from a node of the bottom row to a node on the top row.

• •

Each pair of nodes associated with the same $R$-factor is collapsed into a single node, keeping all arcs fastened to their nodes. Loops appear if the $R$-factors were already contracted (Ricci tensors).

• •

Labels are erased, meaning that the left-to-right reading orders of the product of the $R$ are all equivalent reflecting the usual commutative law for multiplications. The multiplicity may be recovered by examining the Automorphism Group of the new graph

This transformation of Bogdanos’ bipartite graphs on $2n$ nodes to our 2-regular digraphs on $n$ nodes is lossless (reversible).

IV Line Graphs

Each digraph has an associated line digraph which is defined as follows Harary and Norman (1960); Sysło (1982): Each arc in the digraph is rendered as one node in the line digraph. An arc from node $a$ in the line digraph (corresponding to an arc from node $u$ to node $v$ in the digraph) to node $b$ in the line digraph (corresponding to an arc from node $w$ to node $x$ in the digraph) exists if the nodes $v$ and $w$ are the same, which means, if the two arcs in the graph are adjacent to each other and have aligned directions.

Loops in the digraph create loops in the line digraph. A $k$-regular digraph on $n$ nodes preduces a $k$-regular line digraph on $kn$ nodes.

V Single Edge Cuts

If a single arc of the 2-regular digraphs discussed above is cut, a pair of two additional nodes appears at that place: one with indegree 1 and outdegree 0, another with indegree 0 and outdegree 1; the other nodes keep their degrees. The equivalent action in the tensor product is to leave one pair of indices uncontracted; the equivalent action in a $\varphi^{4}$-theory is to move from the vacuum diagrams to the diagrams of Green’s functions. In a preparational step one might have enumerated the marked line graphs where the marked nodes in the line graphs would indicate which lines are cut in their base graphs.

The notation $U_{k,+2}(n,c)$ will be used for the number the unlabeled digraphs with $c$ components with $n-2$ nodes with indegree and outdegree $k$ plus 2 nodes with the mixed in-outdegrees of (0,1) and (1,0) created by the cut. (Adopting the nomenclature of Chae et al. Chae et al. (2007), these “almost” regular graphs will be called fairly regular graphs.) The enumeration for any number of components is $U_{k,+2}(n)\equiv\sum_{c}U_{k,+1}(n,c)$. Only the connected graphs (where $c=1$) are relevant because the graphs with more than one component can be constructed by taking one connected graph and adding any number of graphs of the $k$-regular class. So the ordinary generating function of the fairly regular digraphs $U_{k,+2}(n)$ is the product of the ordinary generating function of $U_{k,+2}(n,1)$ by the ordinary generating function of $U_{k}(n)$, and the equivalent statement holds for the exponential generating function of the labeled graphs.

The full set of these 1-in 1-out graphs up to 6 nodes obtained by cutting an arc follows. The cases without multiarcs up to 7 nodes I have already shown in a manuscript in (Foundation Inc., 2023, A005642).

V.1 1 connected graph on 2 nodes

V.2 1 connected graph on 3 nodes

V.3 3 connected graphs on 4 nodes

V.4 12 connected graphs on 5 nodes

V.5 61 connected graphs on 6 nodes

A summary of the enumeration with one arc cut is in Table 2.

VI Pairs of Cuts

Cutting two distinct arcs of a connected 2-regular digraph produces $U_{2,+4}(n,c)$ unlabeled digraphs with $c$ components where 4 nodes appear at the cuts: 2 with indegree 1 and outdegree 0, and another 2 with indegree 0 and outdegree 1. The $n-4$ “internal” nodes keep their degrees. (Cutting the same arc twice is not interesting because that merely produces two disconnected graphs already covered by the previous section.) The operation with two cuts is topologically more complex than products of a single cut. (i) There is no longer a 1-to-1 correspondence from the graph with the 2-in 2-out nodes to the uncut parent. There are generally two variants of merging the nodes with total degree 1 to rebuild a 2-regular digraph; exceptions are cases with high symmetry like in Section VI.1 which obviously has the parent shown in Section II.1. (ii) If the two cuts target a pair of arcs between a single pair of nodes or more generally two arcs of a 2-connected graph, a graph with more than one component (i.e. two graphs of the type in Section V) may result. So the generating function of $\sum_{c}U_{2,+4}(n,c)$ is the product of the generating functions $U_{2,+4}(n,1)$ and $\sum_{c}U_{2}(n,c)$ to account for graphs where all four nodes of degree 1 are in the same component, plus the symmetrized square generating function of $\sum_{c}U_{2,2}(n,c)$ by $\sum_{c}U_{2}(n,c)$ to account for the graphs where two nodes of degree 1 are in one component, the other two nodes in another component, and any other number of 2-regular graphs in further components. [The symmetrized square of a generating function $g(x)$ is $[g(x)^{2}+g(x^{2})]/2$ following from the cycle index for the symmetric group $S_{2}$ (Flajolet and Sedgewick, 2009, I60).]

A gallery of these 2-in 2-out graphs up to 7 nodes illustrates that concept in the next 3 subsections, showing only the connected digraphs.

VI.1 1 connected graph on 5 nodes

VI.2 4 connected graphs on 6 nodes

VI.3 21 connected graphs on 7 nodes

A summary of the enumeration with two arcs cut is in Table 3.

VII Summary

Table 4 summarizes the bare counts $U_{2}(n,c)$.

The row sums 1, 3, 8, 25, …count the graphs with any number of components. $U_{2}(4)$ and $U_{2}(5)$ have already been computed by Briggs Briggs (1997a, b). We observe that Briggs’ extrapolations to more than 5 nodes Briggs (2000) underestimate the true number of graphs for 6 – 9 nodes.

Appendix A Machine Readable Tables

The ancillary directory contains the information of the unlabeled 2-regular digraphs in files named Reg$n$.txt.zip, where $n$ is the number of nodes. After unzipping, each file contains up to three successive lines per graph:

An arc list for $2n$ arcs, referring to a representative of the labeled graphs created by the $\cal A$-group, with labels from [math] up to $n-1$, in square brackets. Each bracket contains a pair of numbers, separated by a comma; the first is the tail node and second the head node of the arc. Multiarcs are rendered by repeating such pairs. 2. 2.

A capital V, the number of multiarcs in the graph (i.e., the number of entries larger than 1 in the Adjacency Matrix), a blank, the trace of the Adjacency Matrix (i.e., the number of loops), a blank, and the Pólya Cycle Index (a multinomial in the free variables $t_{i}$).

Filtering the lines that start with V0 in these files we obtain 1, 3, 8, 27,…unlabeled, not necessarily connected, 2-regular digraphs with $n\geq 2$ nodes without multiarcs (Foundation Inc., 2023, A005641).

Filtering the lines that start with V0␣0 in these files we obtain 1, 2, 5, 23,…unlabeled, loopless, not necessarily connected, 2-regular digraphs with $n\geq 3$ nodes without multiedges (Foundation Inc., 2023, A219889).

As a further check, filtering the lines that contain ␣0 we obtain 1, 2, 6, 15, 68,…graphs on $n\geq 2$ nodes without loops (which may have multiarcs) (Foundation Inc., 2023, A307180).

One application of this information yields the $r$-rooted unlabeled 2-regular digraphs by defining the generating function $r(x)=1+x$ for the number of ways of labeling a node as 0 (not marked) or 1 (marked), and then substituting $t_{i}\to r(x^{i})$ in the cycle indices. Summation over the cycle indices of all graphs of fixed $n$ generates Table 5.

Another information drawn from the cycle indices are the parities of the permutations of the node-labeled 2-regular digraphs that leave the unlabeled graph intact. A permutation is odd, iff there is an odd number of even-length cycles, i.e., iff for even index $i$ there is an odd exponent $j$ in the factor $t_{i}^{j}$ of the cycle index. For products of different $i$ the product rule for the signs of cycles rules. Example: one of the 25 graphs on 4 nodes has the cycle index $(t_{1}^{4}+2t_{1}^{2}t_{2}+3t_{2}^{2}+2t_{4})/8$.

• •

The term $t_{1}^{4}$ is one permutation with no even index, so this is an even permutation.

• •

The term $2t_{1}^{2}t_{2}$ is two permutations with an even index with exponent 1, so this represents two odd permutations.

• •

The term $3t_{2}^{2}$ is three permutations with an even index with exponent 2, so this represents three even permutations.

• •

The term $2t_{4}$ is two permutations with an even index with exponent 1, so this represents two odd permutations.

The ancillary directory contains the information of the connected unlabeled fairly 2-regular digraphs of Section V in files named Regf$n$.txt.zip, where $n$ is the number of nodes, including the 2 nodes with degree 1. The arc lists contain $2n-3$ arcs and the lines starting with $V$ are the cycle indices.

The ancillary directory contains the information of the connected unlabeled fairly 2-regular digraphs of Section VI in files named RegF$n$.txt.zip, where $n$ is the number of nodes, including the 4 nodes with degree 1. The arc lists contain $2n-6$ arcs and the lines starting with $V$ are the cycle indices.

Appendix B Java Code

The ancillary directory contains the source code of the Java classes in the org/nevec/rjm subdirectory. The javac command in the Makefile compiles the code. The other lines in the Makefile demonstrate how the counts for the unlabeled digraphs are obtained by calling two main classes in succession. The first is Regul which generates incidence matrices for digraphs of mixed connectivity and of specified degree squences and prints the corresponding edge/arc lists which is redirected to an intermediate file. Unlbl2RegGraphHashSet then reads such an edge/arc list and reduces these sets to unlabeled graphs edge lists by running through all row-column permutations and leaving only one representative in its standard output. The two man-pages Regul.1 and Unlbl2RegGraphHashSet.1 provide details of options and arguments of the two classes.

The examples in the Makefile are not optimized for speed. They construct the $L_{2,+2}(n,c)$ for example with 2 nodes of degree 1 and $n-2$ nodes of in- and outdegree 2. Just two geometries occur: the 2 nodes are connected to the same node or to different nodes. So one could gather all cases by running the program twice forgetting the 2 leaf nodes, constructing graphs with $n-2$ nodes: (i) a run with 1 node of indegree 1 and outdegree 1 and n-3 nodes indegree and outdegree 2, (ii) a run with 1 node of indegree 1 and outdegree 2, 1 node of outdegree 1 and indegree 2, and n-4 nodes of indegree and outdegree 2. After the reduction to unlabeled graphs the 2 leaf nodes could be re-attached at the nodes which are uniquely characterized by their underrun of degrees.