# On signed edge domination numbers of graphs

```Signed Edge Domination Numbers in Complete Tripartite Graphs
Dana Collins, Tessa Gromoll. and Daniel Hartman
Department of Mathematics, The University of West Georgia
UWise (University of West Georgia Institutional STEM Excellence) Student Research Program
Abstract
A complete tripartite graph is a graph with three sets of vertices, such that the vertices in each set are connected to all of the other vertices in all of the other sets by edges. In addition, no two vertices in the same set
are connected. The closed neighborhood of an edge is all of the edges which have one vertex in common with that edge. The weight of a subset of edges is the sum of all of the labels of all of the edges in that
subset. A labeling of edges such that the weight of each neighborhood of each edge in the graph is greater than or equal to one is called a signed edge dominating function. The weight of a signed edge dominating
function of a graph is the sum of all of the labels of all the edges of the graph.
In this project, we investigate how to find the minimum weight of a signed edge dominating function of a tripartite graph; which is called the signed edge domination number of that graph. In order to carry out this
research we have explored graphic sequences as a tool for counting such weights. As a result of this research, we have generalized several lemma’s regarding the signed edge domination numbers in complete
tripartite graphs.
Introduction
A complete graph is a graph in
which each vertex in each set is
connected to every other vertex in all of
the other sets by an edge. A tripartite
graph is a graph with three sets of
vertices. Each edge on the graph can be
labeled either as either a positive or
negative weight of one. The weight of a
vertex on a graph is the sum of all of the
weights of the edges connected to that
vertex.
The weight of the entire graph is
the sum of all of the weights of the edges.
The purpose of this research is to
generalize the optimal labeling (known as
the signed edge dominating function) of
each edge, such that the weight of the
graph is at its minimal (known as the
signed edge domination number) and
the closed neighborhood of every edge on
a graph must be greater than or equal to
one.
For our research we let m, n, and p
be the number of vertices in each set. The
graph denoted as Km,n,p where m ≤ n ≤ p,
and m+n ≤ p. To carry out our research,
we examined cases where m, n, and p
can be even or odd integers.
Methods
Consider Lemma 1, where m, n, and p are three
positive even integers such that
m + n ≤ p.
Below is an example of a complete tripartite graph
K2,2,4 , where m = 2, n = 2, and p = 4. The red
edges represent a -1 labeling, and the blue edges
represent a +1 labeling. The weight of each vertex
is the sum of the labeling of the connected edges.
One way to compute the weight of the graph is to
add up the weight of the vertices and divide that
number by two. Notice the weight of this graph is
4, which is the minimal weight for a K2,2,4 graph;
also known as the signed edge domination
number of the graph (w(f)).
Results
Future Studies
There are many ways to continue this research.
Some interesting topics include:
•Further research can be carried out on
cases where m+n ≥ p.
• Also generalizing signed edge domination
numbers for complete n-partite graphs.
References
•S. Akbari, S. Bolouki, P. Hatamib and M. Siami,
On the signed edge domination number of
graphs, Discrete Mathematics 309 (2009), 587594.
•A. Carney and A. Khodkar, Signed edge kdomination numbers in graphs, Bulletin of the
Institute of Combinatorics and its Applications
62 (2011), 66-78.
•B. Xu, On signed edge domination numbers of
graphs, Discrete Mathematics 239 (2001), 179189.
Consider Lemma 2, where m, n, p are three
positive odd integers such that m + n ≤ p.
Below is an example of a complete tripartite graph
K3,3,7, where m = 3, n = 3, and p = 7. Notice the
weight of this graph is 3+3+1 =7, which is the
minimal weight for a K3,3,7 graph.
Acknowledgements
We would like to acknowledge the following for
their help in making this research possible:
• Dr. Abdollah Khodkar
• Dr. S. Swamy Mruthinti
• Dr. Scott Sykes
• This research was made possible by the
funding of the UWISE Program
```