Integer sequence discovery from small graphs

TL;DR

This paper exhaustively analyzes small graphs to discover and verify integer sequences based on graph invariants, contributing new sequences and relationships to the OEIS with an open-source tool.

Contribution

It introduces a comprehensive framework for enumerating small graphs, computing invariants, and discovering new integer sequences, with tools for visualization and extensibility.

Findings

141 new integer sequences added to OEIS

6 sequences corrected or appended

Programmatic relationships among invariants identified

Abstract

We have exhaustively enumerated all simple, connected graphs of a finite order and have computed a selection of invariants over this set. Integer sequences were constructed from these invariants and checked against the Online Encyclopedia of Integer Sequences (OEIS). 141 new sequences were added and 6 sequences were appended or corrected. From the graph database, we were able to programmatically suggest relationships among the invariants. It will be shown that we can readily visualize any sequence of graphs with a given criteria. The code has been released as an open-source framework for further analysis and the database was constructed to be extensible to invariants not considered in this work.