# The Enumeration of Graph Imbeddings

8-1990

## Degree Name

Doctor of Philosophy

Mathematics

## Abstract

Graphs can be drawn on surfaces. Here, graphs may have multiple edges or loops (pseudographs), the surfaces are closed orientable 2-manifolds (sphere, torus, etc.) and their generalizations (pseudosurfaces and generalized pseudosurfaces), and the drawings are 2-cell imbeddings. For quite some time it has been known that a connected graph has $\rm \Pi(degree(\upsilon)-1)$! 2-cell imbeddings on surfaces. More detailed information about these imbeddings has been wanting. In addition, many of the imbeddings counted above 'look the same' when all vertex and edge labels are removed. The resulting unlabeled imbeddings are fewer in number and more difficult to enumerate than their labeled counterparts.

This dissertation enumerates the labeled and unlabeled 2-cell imbeddings of two important classes of pseudographs, bouquets (one vertex, many loops) and dipoles (two vertices, multiple edges), on surfaces, pseudosurfaces, and generalized pseudosurfaces. The enumeration is by size, by size and number of regions, by size and region distribution ($r\sb{k}$ regions of boundary length k), and, in the case of the bouquet, by symmetry. For example, the fraction of all labeled 2-cell imbeddings of the dipole with n edges having r regions on surfaces is twice the fraction of permutations in the symmetric group of degree n + 1 having r orbits provided n and r have the same parity.

A correspondence between imbeddings of the bouquet on pseudosurfaces and the totality of all pseudograph imbeddings on surfaces is exploited to enumerate the latter in both the labeled and unlabeled case. For example, there are four connected pseudographs with two edges; this set of pseudographs has a total of twenty labeled and five unlabeled 2-cell imbeddings on surfaces.

## Access Setting

Dissertation-Open Access

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