Counting five-node subgraphs

TL;DR

This paper derives exact formulas for counting all non-induced connected subgraphs of five nodes in simple graphs, providing a combinatorial approach applicable to larger subgraphs and demonstrating applications on real-world networks.

Contribution

It introduces exact count formulas for five-node subgraphs and offers a combinatorial proof method adaptable to larger subgraphs, advancing subgraph enumeration techniques.

Findings

Exact formulas for 21 five-node subgraphs derived

Application demonstrated on regular and real-world networks

Linear relations between induced and non-induced subgraph counts discussed

Abstract

We propose exact count formulae for the 21 topologically distinct non-induced connected subgraphs on five nodes, in simple, unweighted and undirected graphs. We prove the main result using short and purely combinatorial arguments that can be adapted to derive count formulae for larger subgraphs. To illustrate, we give analytic results for some regular graphs, and present a short empirical application on real-world network data. We also discuss the well-known result that induced subgraph counts follow as linear combinations of non-induced counts.