English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A gain graph is a graph whose edges are orientably labelled from a group. Aweighted gain graph is a gain graph with vertex weights from an abeliansemigroup, where the gain group is lattice ordered and acts on the weightsemigroup. For weighted gain graphs we establish basic properties and wepresent general dichromatic and forest-expansion polynomials that are Tutteinvariants (they satisfy Tutte's deletion-contraction and multiplicativeidentities). Our dichromatic polynomial includes the classical graph one byTutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs withpositive integer weights, and that of rooted integral gain graphs by Forge andZaslavsky. It is not a universal Tutte invariant of weighted gain graphs; thatremains to be found.  An evaluation of one example of our polynomial counts proper list colorationsof the gain graph from a color set with a gain-group action. When the gaingroup is Z^d, the lists are order ideals in the integer lattice Z^d, and thereare specified upper bounds on the colors, then there is a formula for thenumber of bounded proper colorations that is a piecewise polynomial function ofthe upper bounds, of degree nd where n is the order of the graph.  This example leads to graph-theoretical formulas for the number of integerlattice points in an orthotope but outside a finite number of affinographichyperplanes, and for the number of n x d integral matrices that lie between twospecified matrices but not in any of certain subspaces defined by simple rowequations.

Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs