On the Genus Distribution of $(p,q,n)$-Dipoles

Abstract

There are many applications of the enumeration of maps in surfaces to other areas of mathematics and the physical sciences. In particular, in quantum field theory and string theory, there are many examples of occasions where it is necessary to sum over all the Feynman graphs of a certain type. In a recent paper of Constable et al. on pp-wave string interactions, they must sum over a class of Feynman graphs which are equivalent to what we call $(p,q,n)$-dipoles. In this paper we perform a combinatorial analysis that gives an exact formula for the number of $(p,q,n)$-dipoles in the torus (genus 1) and double torus (genus 2).

Genus distributions of graphs under edge-amalgamations

“…Prior research on counting imbeddings on various orientable and non-orientable surfaces includes [3], [4], [5], [10], [11], [13], [16], [17], [18], [19], [21], [23], [24], [25], [26], [27], [28], [29], [30], and [31]. Prior work on counting graph imbeddings in a minimum-genus surface includes [2], [7], [6], and [15].…”

Total embedding distributions of Ringel ladders

“…Gross in 1980s. Since then, it has been attracted a lot of attentions, for the details, we may refer to [1,8,9,10,11,13,16,17,19,22,26,28,31,32,33,34,35,36,37,38] etc (We only list a few). However, for the total embedding distributions, only few classes are known.…”

Genus Distributions of 4-Regular Outerplanar Graphs

“…Prior work on genus distributions has been largely focused on graph families with high symmetries. In this context, prior work on counting embeddings in all orientable surfaces or in all surfaces includes [ChLiWa06], [KwLe93], [KwLe94], [KwSh02], [McG87], [Mu99], [St90], [St91a], [St91b], [Tesa00], [ViWi07], [WaLi06], and [WaLi08]. The well known Heffter-Edmonds algorithm calculates the genus distribution of any graph, but its time-complexity is superexponential in the size of the graph (see [GrTu87]).…”