# 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.

## Key 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.