$3$-Connected Simple Graphs with The Same Genus Distributions

Abstract

The paper construct infinite classes of non-isomorphic $3$-connected simple graphs with the same total genus polynomial, using overlap matrix, symmetry and Gustin representation. This answers a problem (Problem $3$ of Page $38$) of L.A. McGeoch in his PHD thesis.
The result is helpful for firms to make marketing decisions by calculating the graphs of user demand relationships of different complex ecosystems of platform products and comparing genus polynomials.