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Keywords:

  • graph embedding;
  • total embedding distribution;
  • circular ladders;
  • overlap matrix;
  • Chebyshev polynomials

Abstract

The total embedding polynomial of a graph G is the bivariate polynomial

  • display math

where inline image is the number of embeddings, for inline image into the orientable surface inline image, and inline image is the number of embeddings, for inline image into the nonorientable surface inline image. The sequence inline image is called the total embedding distribution of the graph G; it is known for relatively few classes of graphs, compared to the genus distribution inline image. The circular ladder graph inline image is the Cartesian product inline image 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.