Introducing a vertex polynomial invariant for embedded graphs

TL;DR

This paper introduces a new vertex polynomial invariant for embedded graphs, connecting it to existing polynomials and topological concepts, and providing recursive and topological interpretations.

Contribution

It defines the vertex polynomial invariant, resolves a vertex count problem, and links it to topological and combinatorial graph polynomials.

Findings

Vertex polynomial quantifies vertex distribution under twisted duality.

Polynomial depends only on signed intersection graphs for bouquets.

Provides recursive relations and topological interpretations for the polynomial.

Abstract

The ribbon group action extends geometric duality and Petrie duality by defining two embedded graphs as twisted duals precisely when they lie within the same orbit under this group action. Twisted duality yields numerous novel properties of fundamental graph polynomials. In this paper, we resolve a problem raised by Ellis-Monaghan and Moffatt [Trans. Amer. Math. Soc. 364 (2012), 1529--1569] for vertex counts by introducing the vertex polynomial: a generating function quantifying vertex distribution across orbits under the ribbon group action. We establish its equivalence via transformations of boundary component enumeration and derive recursive relations through edge deletion, contraction, and twisted contraction. For bouquets, we prove the polynomial depends only on signed intersection graphs. Finally, we provide topological interpretations for the vertex polynomial by connecting this polynomial to the interlace polynomial and the topological transition polynomial.