#### Date of Defense

Spring 4-12-2004

#### Department

Mathematics

#### First Advisor

Ping Zhang, Mathematics

#### Second Advisor

Paul Eenigenburg, Mathematics

#### Third Advisor

Christine Browning, Mathematics

#### Abstract

Let D be an oriented graph of order *n* and size *m*. A Ɣ-labeling of *D* is a one-to-one function f' : *V*(*D*) → {0,1,2,...,*m*} that induces a labeling *f'* : *E*(*D*) → {±1,±2,...,±*m*} of the arcs of *D* defined by *f'*(*e*) = *f*(*v*) - f(*u*) for each arc *e* = (*u,v*) of *D*. The value of a Ɣ-labeling *f* is val(*f*) = ∑_{e∈E(G)}*f*'(*e*). A Ɣ-labeling of *D* is balanced if the value of* f* is 0. A Ɣ-labeling of *D* graceful if the induced edge-labeling *f'* is also one-to-one. An oriented graph *D* is balanced if *D* has a balanced labeling. A graph *G* is orientably balanced if *G* has a balanced orientation. In this project, we study the properties of balanced Ɣ-labelings of oriented graphs and orientably balanced graphs.

#### Recommended Citation

Herlein, Scott, "On Ɣ-Labeling of Oriented Graphs" (2004). *Honors Theses*. 273.

http://scholarworks.wmich.edu/honors_theses/273

#### Access Setting

Honors Thesis-Campus Only