Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs

TL;DR

This paper introduces efficient algorithms for computing chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs, providing new formulas and duality insights for these graph classes.

Contribution

It presents a polynomial-time algorithm for chromatic polynomials of generalized vertex joins of trees and derives flow polynomials for outerplanar graphs via duality.

Findings

Polynomial-time algorithm for chromatic polynomials of generalized vertex joins of trees

Closed-form formulas for chromatic and flow polynomials of generalized wheel graphs

Duality-based computation of flow polynomials for outerplanar graphs

Abstract

A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees, and by duality we find the flow polynomials of arbitrary outerplanar graphs. We also present closed formulas for the chromatic and flow polynomials of vertex joins of cliques and cycles, otherwise known as "generalized wheel" graphs.