Genus Polynomials for Three Types of Graphs

Abstract

On the basis of joint trees introduced by Yanpei Liu, by choosing different spanning trees and classifying the associated surfaces, we obtain the explicit expressions of genus polynomials for three types of graphs, namely $K_5^n,W_6^n$ and $K_{3,3}^n$, which are different from the graphs whose embedding distributions by genus have been obtained. And $K_5^n$ and $K_{3,3}^n$ are non-planar.