Total embedding distributions of circular ladders

The total embedding polynomial of a graph G is the bivariate polynomial IG(x,y)=Σi=0∞aixi+Σj=1∞bjyj,where ai is the number of embeddings, for i=0,1,., into the orientable surface Si, and bj is the number of embeddings, for j=1,2,., into the nonorientable surface Nj. The sequence {ai(G)|i≥0} {bj(G)|j≥1} is called the total embedding distribution of the graph G; it is known for relatively few classes of graphs, compared to the genus distribution {ai(G)|i≥0}. The circular ladder graph CLn is the Cartesian product K2□Cn of the complete graph on two vertices and the cycle graph on n vertices. In this article, we derive a closed formula for the total embedding distribution of circular ladders.