||Competition Graph Theory
Let D be a directed graph on n vertices.
We define a competition graph as an undirected graph with the same vertex set as D where ther is an edge between vertices x and y if there exists a vertex u in V(D) such that [u,x] and [u,y] in E(D).
We consider the vertices x and y in competition since they are receiving something from the same source, namely vertex u.
Competition graphs have applications in modeling ecosystems, radio transmission networks and energy grids.
- Week 1:
- During the initial week I pored over some of the papers that Dr. Fiorini had sent to me by e-mail.
In this process I reviewed basic concepts and definitions of graph theory such as paths, cycles, vertex degree, graph isomorphisms etc.
The contents of the papers focused on the topic of competition graph theory and open problems associated with it.
- Week 2:
- On Tuesday I met with my mentor Dr. Fiorini to summarize the materials I reviewed and began assigning some tasks.
First he wanted a code in Matlab that could generate a digraph and the associated competition graph given a criteria on the arc connectivity.
During Wednesday and Thursday I worked on and completed the code.
After running several examples I began to notice certain graph types that did not appear and shared this with my mentor.
From there he charged me with writing up the proofs of these conjectures if they held.
- Week 3:
- I continued work on summarizing articles by finding articles related to some of the articles I had already reviewed.
I also continued work on the proof of my conjecture though I have been stuck on writing some of the details.
- Week 4:
- I finally made somewhat of a breakthrough this week when I completed a proof of my conjecture.
After sharing this with one of my partners he inspired me to work on another conjecture which I proved.
After presenting my results to my mentor and Dr. Nakamura he tasks us with making our conjectures broader to include more cases and to cover different types of examples.
- Week :
- In the pursuit of generalising the proof and expanding on results I became stuck on some of the details.
In a group session Dr. Fiorini pointed out some of my errors in logic and loose ends.
He gave me a new direction to work in which led to more fruitful results.
I plan next week to meet with Dr. Fiorini to tighten up the proof.
Also he gave back some of our previously worked on materials with corrections and suggestions that I have addressed in part.