Abstract:For a simple undirected graph $G$, we derive exact formulae for the number of copies of each of the 21 non-isomorphic connected graphs on five vertices. For nine of these graphs (the stingray, spinning top, kite, ufo, crown, envelope, lamp, arrowhead, and cat's cradle) the resulting analytical formulae appear to provide substantially new formulations; we also identify and correct five erroneous published formulae for the 5-path, banner, and lollipop. The proofs use elementary combinatorial arguments, organized around recurring constructions based on incident structures, walks, common neighbourhoods, and neighbourhood subgraphs, several of which extend naturally to larger subgraphs. We illustrate the formulae by deriving analytical results for several regular graphs and by applying them to a real-world network, where induced five-node subgraph counts, obtained as linear combinations of general counts, are compared with Erdős-Rényi and degree-preserving null ensembles.
From: Steve Lawford [view email]
[v1]
Wed, 23 Sep 2020 18:00:45 UTC (1,373 KB)
[v2]
Thu, 17 Sep 2026 14:45:56 UTC (1,700 KB)
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