Lattice points in orthotopes and a huge polynomial Tutte invariant of weighted gain graphs

A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte’s deletion-contraction and multiplicative identities). Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky’s two for gain graphs, Noble and Welsh’s for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that remains to be found. An evaluation of one example of our polynomial counts proper list colorations of the gain graph from a color set with a gain-group action. When the gain group is Zd, the lists are order ideals in the integer lattice Zd, and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function, of degree d|V |, of the upper bounds. This example leads to graph-theoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n × d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations. Mathematics Subject Classifications (2010): Primary 05C22; Secondary 05C15, 05C31, 52B20, 52C35. Part of the research of this author was performed while visiting the State University of New York at Binghamton in 2005. Part of his research was supported by the TEOMATRO project, grant number ANR10-BLAN 0207, in 2013. Some of the work of this author was performed while visiting the Laboratoire de recherche en informatique, Université Paris-Sud, Orsay, in 2007. 1