On terminal forms for topological polynomials for ribbon graphs: The $N$-petal flower

TL;DR

This paper extends the understanding of the Bollobas-Riordan polynomial for ribbon graphs by explicitly computing it for certain terminal forms, specifically N-petal rosette graphs, thereby broadening the class of graphs with known polynomial formulas.

Contribution

It provides explicit Bollobas-Riordan polynomial formulas for N-petal rosette ribbon graphs, expanding the set of terminal forms for which the polynomial can be directly computed.

Findings

Derived Bollobas-Riordan polynomial for N-petal rosette graphs.

Expanded the class of terminal forms with known polynomial expressions.

Enhanced computational methods for ribbon graph invariants.

Abstract

The Bollobas-Riordan polynomial [Math. Ann. 323, 81 (2002)] extends the Tutte polynomial and its contraction/deletion rule for ordinary graphs to ribbon graphs. Given a ribbon graph $\cG$, the related polynomial should be computable from the knowledge of the terminal forms of $\cG$ namely specific induced graphs for which the contraction/deletion procedure becomes more involved. We consider some classes of terminal forms as rosette ribbon graphs with $N\ge 1$ petals and solve their associate Bollobas-Riordan polynomial. This work therefore enlarges the list of terminal forms for ribbon graphs for which the Bollobas-Riordan polynomial could be directly deduced.