# Alvaro Belmonte ’20

## "K-Super Bondage Number on Graphs"

Major: Math
Hometown: Bethlehem, PA

We studied the reliability of a network, based upon failure or deletions of edges, and with the criteria of failure being the dominant number. The Dominant number is the size of the smallest set of vertices such that every vertex is adjacent to one in the set or is in the set itself. The bondage number is the least number of edges deleted from a graoh such that the dominant number of the graph could increase. K-Super Bondage number is a natural extension of the bondage number of a graph, where instead of just asking increase, we say that is the least amount of edges deleted such that the dominant number could increase by k.

### Describe the origin of your project.

We wanted to work with network reliability. For that, we needed to choose what was going to be our criteria for failure and what was going to fail or be deleted. My partner gave the idea to work with dominant numbers, with other teammates offering other ideas. We decided to work we deleting edges because it was more consistent according to the literature. We read that deletion of edges may either increase the dominant number or just leave it how it is, meanwhile deleting vertices could increase, decrease, or maintained constant.

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

He helped us in instances where higher knowledge of some sort was needed by explaining it to us in simple terms. HE also was helpful in polishing proofs by giving us logic’s pitfalls on the drafts and just discussing the idea pf the proof.

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

My English has always been one of the biggest obstacles. I come from Venezuela, so I learned English as my second language five years ago when I moved here. This is one of my biggest obstacles because I sometimes cannot explain my ideas clearly and concisely as I would like, which is a strong thing to have when you are working in a group research project.

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

My biggest take away has been my experience doing research as a career, with every other skill that must come with that. I also will be finishing a paper that our group would want to submit to a math journal before my graduate applications to give me a boost for such. This experience has settled that I want to go to graduate studies for my Ph.D. in Mathematics

### What was the result of your project?

We found the k-Super Bondage number of some families of graphs, depending on how many vertices it has. Some general bounds for such. We also found the Minimal k-super bondage’s graphs from the set of random graphs, G(n,m). We have the beginning of data to generalize the maximal k-super bondage’s graphs from the set of random graphs.

### In your own words, how do you feel about being awarded this opportunity? Why should other students take advantage of the SOAR program at Moravian College?

I feel proud, accomplished and grateful for being awarded this opportunity. For students who are insecure about going to graduate studies, SOAR is the right thing to take advantage of to understand how research is going to be.

### Now that SOAR is over, do you plan to expand upon your research? If so, how?

Yes, because I am still in contact with my teammates for the program to keep writing the paper for submission, and to expand on stuff we did not work or add other characteristics to the graph, such that the problem may be harder but we would already know the basics.

### Have you, or do you plan to present this research outside the SOAR presentations?

Yes, I plan to present in two conferences this fall:

• 8th Annual Regional Undergraduate Mathematics Research Conference at Towson University (near Baltimore, Maryland) On November 2nd, 2019.
• EPaDel Fall 2019 Section Meeting On November 9th, 2019.

For the springtime, I do not know yet because I am also doing an honor’s project so I would not know which to present in the Moravian student math conference.