### General Information

Student: Alan Jara CoRE 444 New York City College of Technology alanjara@reu.dimacs.rutgers.edu Applying Graph Pebbling to Resource Allocation During Extreme Events

### Project Description

Global Climate Change has caused an increase in natural disasters such as hurricanes, coastal floods, tornadoes, wildfires, etc. Natural disasters almost always and immediately create difficulties in delivering aid and resources to the effected regions. This could be expensive given that you had to take alternate routes to an area that's close by. However, an increase in natural disasters has also given rise to more efficient ways to transport aid. My research focuses on transporting/moving resources from one region to the other while still maintaining a cost effective threshold.

### Weekly Log

Week 1:
During the first week, I have met most of the faculty and my group project members as well as my mentors. The students I am working with are Adam Ibrahim and Kistine Andall. We made plans and constructed a powerpoint presentation explaining what our research project will be on in addition to some basics on Graph Theory and Graph Pebbling. We are currently doing some literature review to have a better understanding of Graph Theory and Graph Pebbling in order to come up with some concrete ideas on how to approach our project.
Week 2:
We got some reading done and familiarized ourselves with the basics of Graph Theory and Graph Pebbling. Additionally, we summarized what we have read and are brainstorming for different ideas.
Week 3:
Currently, we are working on a formal proof by induction of the formula for the pebbling number for Paths that is; Π(Pn) = 2n-1. We are also looking to see if we could find a general formula for the pebbling number for a weighted Path.
Week 4:
On week 4, my colleagues and I were studying Kruskals Algorithm, Dijkstra's Algorithm, and Breadth-First-Search. These algorithms are mainly used to find the shortest path and a spanning tree of a graph from an arbitrary root vertex.
Week 5:
During week 5 we started to brainstorm on how to implement a pebbling move into an adjacency matrix and a transitional matrix for some time t.
Week 6:
My mentor has given us a suggestion on how to implement a pebbling move into an adjacent transitional matrix. With this we are trying to simplify and extend her matrix. She also made me go through some proofs for eigenvalues and diagonal entries for non-negative symmetric matrices.