1. Minimum Pk-total weight of simple connected graphs
Jeremy Holden
Norwich University
Mentor:
Dr. Ji Young Choi
Shippensburg University

2. Graph
A graph is simply a collection of vertices and edges. An edge connects two vertices together. The purpose of a graph is to represent a structure of connections between nodes. Example:

3. Simple connected graphs
Simple, not connected.
Connected, not simple.
Simple and connected.
Definition:
A simple graph must not contain loops, and only one edge may connect two given vertices. The middle graph violates both of these conditions.
In a connected graph, you can always walk from one vertex to another. The left graph violates this condition.

4. k-paths
A k-path is a “walk” through a graph in which you start at a vertex and no edge is repeated, and includes k edges.
G =
3-path
3-path
Not a path. You cannot visit all three edges without repeating an edge.

5. Pk-total weight
To compute the Pk-total weight of a graph, you first assign +1 or -1 to every edge. You can do this however you like.
You then make a list of every distinct k-path and add up the weights of the edges in each path. You then add up all the paths. This number is the Pk-total weight.
We will compute P1 and P2 of this graph. 1 1 1 -1

6. P1-total weight P1-total weight = 1 + 1 + 1 + (-1) = 2 1 1 G = 1 -1 1

7. P2-total weight = 2 + 2 + 0 + 0 + 2 = 6 1 1 G = 1 -1 1 1 1 1 1 -1 1 1

8. The Minimum Pk-total weight is the minimum of the absolute value of the Pk-total weight considering every possible +/-1 edge assignment. It is clear that there are 2^|E| possible edge labelings, where |E| is the number of edges in our graph. Examples: 1 -1 1 -1
P1(G) = 0 1 1
P2(G) = 0 -1 -1

9. 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1
P3(G) =
|(-1) + (-1) + 1| = 1
This is the minimum, since there are three 3-paths, and odd + odd + odd = odd, so you cannot get 0.

10. Problems I am working on:
Can I get an explicit formula for Pk(Kn) for all k and n?
What about explicit formulas for other families?
What properties of a graph G are needed to guarantee that Pk(G) = 0 for all k? Or for all k ≠ 1.
Note: Kn is called the complete graph on n vertices. It is constructed by drawing n vertices, and every vertex is connected to every other vertex. K5 is shown above.

11. Questions?