Author(s)

Michael Hecht^1,2,3, Ivo F. Sbalzarini^1,2,3

¹Technische Universität Dresden, Faculty of Computer Science, Dresden, Germany.
²Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany.
³Center for Systems Biology Dresden, Dresden, Germany.

We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By considering two-dimensional CW complex of elementary cycles and deriving formulas for the Betti numbers of the associated cellular homology groups, we extend the list of representation independent topological inavariants measuring the graph structure. We prove the computation of the 2nd Betti number to be sharp #P hard in general and present specific representation invariant sub-fillings yielding efficiently computable homology groups. Finally, we suggest how to use the provided structural measures to shed new light on graph theoretical problems as graph embeddings, discrete Morse theory and graph clustering.

Biggs Theorem, Elementary and Simple Cycles, CW Complexes of Graphs, Cellular and Singular Homology, Betti Numbers

Hecht, M. and Sbalzarini, I. (2021) Biggs Theorem for Directed Cycles and Topological Invariants of Digraphs. Advances in Pure Mathematics, 11, 573-594. doi: 10.4236/apm.2021.116037.