Total Embedding Distributions of Circular Ladders

The total embedding polynomial of a graph G is the bivariate polynomialI G ( x , y ) = ∑ i = 0 ∞ a i x i+∑ j = 1 ∞ b j y j , where a i is the number of embeddings, for i = 0 , 1 , ... , into the orientable surface S i , and b j is the number of embeddings, for j = 1 , 2 , ... , into the nonorientable surface N j . The sequence{ a i ( G )| i ≥ 0 } ⋃ { b j ( 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{ a i ( G )| i ≥ 0 } . The circular ladder graph C L nis the Cartesian productK 2 □ C nof 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.