# Adam Buzzard '16

Major/Minor: Mathematics
Hometown: Bangor, PA
Project Advisor: Dr. Nathan Shank

## Network Reliability: The k-Neighbor Component Edge Connectivity

### Briefly describe your project.

Graph theory is a category of math that is used to analyze network reliability. Connectivity is one measure of the reliability of a network. There are several different connectivity parameters that have been explored in the past. Dr. Shank and I designed a new connectivity parameter and used the summer to determine exactly how it acts and prove some results.

### Describe the origin of your project.

Spring semester, I did an independent study in extremal graph theory with Dr. Shank. We covered various topics, but I took a real interest in connectivity.  We came up with a new parameter and decided to explore it further in a SOAR project this summer. The project and decision to work with each other was pretty much a joint decision.

### What’s the best part about working with your faculty mentor? What valuable insights have they brought to your project?

I’ve worked with Dr. Shank a lot in the past, and I’ve always enjoyed it.
He’s very knowledgeable and always able to point me in the right direction when I get stuck. He’s also full of good ideas and new ways to look at things when we’re both stumped.

### What has been your biggest obstacle so far?

The biggest obstacle has been writing up the proofs for our conjectures. Mathematical proofs have to be extremely precise. Getting the wording just right is really important. I’ve had to write proofs for several of my math classes, but it’s a lot different when it’s something nobody has done before.

### What has been your biggest takeaway from this experience?

My biggest takeaway has been that the research process takes a really long time. I’ve always set my goals pretty high when it comes to research, but the fact is, it just takes time. It’s better to make sure that when you’re doing is correct than rushing through, trying to prove a bunch of different things, and then realizing you’re wrong. Knowing this, I think I’ll be smarter about the way I approach my honors project this coming year.

### What was the result of your project? Was it congruent with your hypothesis?

We didn’t really have a specific hypothesis. We weren’t sure exactly how this new parameter would act until we applied it to some different graphs. We were able to prove some results on different graphs and wrote a paper that we hope to submit to a few journals.

### Do you think you’ll be able to extend on your research after this summer is over? If so, where would you like to see it go?

I will definitely extend the project this summer! Our new definition of connectivity has proven to be very different than many previous definitions. I will also be facilitating an  honors project next year and hope to prove more results about the k-neighbor component connectivity.