#### Degree Name

MS (Master of Science)

#### Program

Mathematical Sciences

#### Date of Award

12-2015

#### Committee Chair or Co-Chairs

Debra Knisley, Ph.D.

#### Committee Members

Teresa Haynes, Ph.D., Robert Gardner, Ph.D.

#### Abstract

If every vertex in a graph G has the same degree, then the graph is called a regular graph. That is, if deg(v) = r for all vertices in the graph, then it is denoted as an r-regular graph. A graph G is said to be vertex-weighted if all of the vertices are assigned weights. A generalized definition for degree regularity for vertex-weighted graphs can be stated as follows: A vertex-weighted graph is said to be r_{w}-regular if the sum of the weights in the neighborhood of every vertex is r_{w}. If all vertices are assigned the unit weight of 1, then this is equivalent to the definition for r-regular graphs. In this thesis, we determine if a graph has a weighting scheme that makes it a weighted regular graph or prove no such scheme exists for a number of special classes of graphs such as paths, stars, caterpillars, spiders and wheels.

#### Document Type

Thesis - Open Access

#### Recommended Citation

Samani, Franklina, "On Properties of r_{w}-Regular Graphs" (2015). *Electronic Theses and Dissertations.* Paper 2601. https://dc.etsu.edu/etd/2601

#### Copyright

Copyright by the authors.