Orbit-counting polynomials for graphs and codes

We construct an "orbital Tutte polynomial" associated with a dual pair M and M of matrices over a principal ideal domain R and a group G of automorphisms of the row spaces of the matrices. The polynomial has two sequences of variables, each sequence indexed by associate classes of elements of R. In the case where M is the signed vertex-edge incidence matrix of a graph over the ring of integers, the orbital Tutte polynomial specialises to count orbits of G on proper colourings of or on nowhere-zero flows or tensions on with values in an abelian group A. (In the case of flows, for example, we must substitute for the variable xi the number of solutions of the equation ia = 0 in the group A. In particular, unlike the case of counting nowhere-zero flows, the answer depends on the structure of A, not just on its order.) In the case where M is the generator matrix of a linear code over GF(q), the orbital Tutte polynomial specialises to count orbits of G on words of given weight ...

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Peter J. Cameron, Bill Jackson, Jason D. Rudd
• Keywords: DM 2008, Orbital Tutte Polynomial, Principal Ideal Domain, Vertex-edge Incidence Matrix