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

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:

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.

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.

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

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

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

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

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.

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.

Questions?