Date of Award


Degree Name

Doctor of Philosophy



First Advisor

Dr. Arthur T. White

Second Advisor

Dr. Allen J. Schwenk

Third Advisor

Dr. Clifton Ealy Jr.

Fourth Advisor

Dr. Michael Raines


The standard non-Euclidean geometries, hyperbolic geometry and elliptical geometry, both arise by negating the parallel postulate of Euclid. Both these geometries share with Euclidean geometries an infinitude of points and lines. But also possible are many finite geometries. Among these are the class of projective geometries PG(m,q) of projective dimension m (m > 2) and the k-configurations. These mathematical objects, although primarily geometric in nature, provide related structures of combinatorial interest: block designs. These have applications in scheduling problems and the design of experiments for statistical analysis. Recently, A. T. White has added a topological flavor to the study of the geometries PG{m,q). In particular, when m = 2 he found models, as imbeddings of Cayley graphs on surfaces and pseudosurfaces, using voltage graphs (obtained from the classical field construction of PG(m,q)). In this dissertation we have extended his work for the case m + I a prime number in a similar manner. We have also used this approach (and ad hoc methods at times) to successfully provide concise voltage graph constructions for 3-configurations (a -configuration is a finite geometry such that exactly k points lie on any line, exactly m lines intersect at any given point, where m is a constant, and any two points determine at most one line). The class of 3-configurations is interesting since it contains some classical finite geometries, e.g., the Fano plane and the Desargues and Pappus configurations.

Access Setting

Dissertation-Open Access