A Hopf algebra for counting cycles

Simple cycles, also known as self-avoiding polygons, are cycles on graphs which are not allowed to visit any vertex more than once. We present an exact formula for enumerating the simple cycles of any length on any directed graph involving a sum over its induced subgraphs. This result stems from an Hopf algebra, which we construct explicitly, and which provides further means of counting simple cycles. Finally, we obtain a more general theorem asserting that any Lie idempotent can be used to enumerate simple cycles.

[1]  Russell Merris,et al.  Single-hook characters and hamiltonian circuits ∗ , 1983 .

[2]  Gordon G. Cash,et al.  The number of n-cycles in a graph , 2007, Appl. Math. Comput..

[3]  Christian Choffrut,et al.  Determinants and Möbius functions in trace monoids , 1999, Discret. Math..

[4]  Donald B. Johnson,et al.  Finding All the Elementary Circuits of a Directed Graph , 1975, SIAM J. Comput..

[5]  William R. Schmitt,et al.  Antipodes and incidence coalgebras , 1987, J. Comb. Theory, Ser. A.

[6]  Noga Alon,et al.  Finding and counting given length cycles , 1997, Algorithmica.

[7]  Q. Wu,et al.  S. Montgomery, Hopf Algebras and their Actions on Rings, CBMS Regional Conference Series in Mathematics, 82, Providence, RI, 1993. , 2003 .

[8]  Thibault Espinasse,et al.  Relations Between Connected and Self-Avoiding Hikes in Labelled Complete Digraphs , 2016, Graphs and Combinatorics.

[9]  G. Rota.  On the Foundations of Combinatorial Theory , 2009 .

[10]  Andreas Björklund.  Determinant Sums for Undirected Hamiltonicity , 2014, SIAM J. Comput..

[11]  N. Madras,et al.  THE SELF-AVOIDING WALK , 2006 .

[12]  Wilhelm von Waldenfels,et al.  Zur Charakterisierung Liescher Elemente in freien Algebren , 1966 .

[13]  William Schmitt,et al.  Incidence Hopf algebras , 1994 .

[14]  H. Duminil-Copin,et al.  The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$ , 2010, 1007.0575.

[15]  Christophe Reutenauer,et al.  Higher Lie Idempotents , 1999 .

[16]  René Schott,et al.  Complexity of counting cycles using zeons , 2011, Comput. Math. Appl..

[17]  N. P. Khomenko,et al.  Identifying certain types of parts of a graph and computing their number , 1972 .

[18]  Richard M. Karp,et al.  Dynamic programming meets the principle of inclusion and exclusion , 1982, Oper. Res. Lett..

[19]  William Schmitt.  Hopf Algebras and Identities in Free Partially Commutative Monoids , 1990, Theor. Comput. Sci..

[20]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[21]  Pierre Cartier,et al.  Problemes combinatoires de commutation et rearrangements , 1969 .

[22]  John Milnor,et al.  On the Structure of Hopf Algebras , 1965 .

[23]  Paul Rochet,et al.  Algebraic Combinatorics on Trace Monoids: Extending Number Theory to Walks on Graphs , 2016, SIAM J. Discret. Math..

[24]  F. Patras,et al.  Logarithmic derivatives and generalized Dynkin operators , 2012, 1206.4990.

[25]  Volker Diekert.  Transitive Orientations, Möbius Functions, and Complete Semi-Thue Systems for Free Partially Commutative Monoids , 1988, ICALP.

[26]  Eric T. Bax,et al.  Inclusion and Exclusion Algorithm for the Hamiltonian Path Problem , 1993, Inf. Process. Lett..

[27]  Russell Merris.  Immanantal invariants of graphs , 2005 .

[28]  Christophe Reutenauer,et al.  On Dynkin and Klyachko Idempotents in Graded Bialgebras , 2002, Adv. Appl. Math..

[29]  Joel Franklin,et al.  A Finite-Difference Sieve to Count Paths and Cycles by Length , 1996, Inf. Process. Lett..

[30]  M. Lapidus,et al.  Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot , 2004 .

[31]  Victor Reiner,et al.  Hopf Algebras in Combinatorics , 2014, 1409.8356.