**Project/blog link:** Cliques and Stones: Relating Minimum Degree to the Optimal Pebbling Number in Graphs**BASIS Advisor:** Ms. Marizza Bailey**Internship location:** ASU, School of Mathematical and Statistical Sciences**Onsite Mentor:** Dr. Andzrej Czygrinow, Associate Professor

A graph is a mathematical object consisting of the abstract notions of vertices and edges, where the vertices represent objects, and the edges connecting certain pairs of vertices may represent any sort of relation between those objects. As such, graph theory allows us to model any imaginable network, giving us the power to illuminate more complex and obscure concepts within fields such as chemistry, biology, informational studies, and artificial intelligence by providing a more structural approach. On campus at Arizona State University, I am currently researching graph theory concepts to apply them to more practical systems. Specifically, my research will look at an area of interest in graph theory called pebbling, which involves the very simple idea of transferring pebbles between vertices by moving them along the edges of a graph. The idea of pebbling on graphs can be used most directly to represent the transportation of resources. For example, the pebbles may represent fuel containers, and as pebbles are moved along edges, some amount of pebbles are lost, signifying the cost of transporting the fuel. By looking at certain properties of graphs and how they relate to the efficiency of moving pebbles along edges in order to reach particular vertices, I hope to be able to discover more about how to effectively construct and manage specific types of networks.

Cliques and Stones: Relating Minimum Degree to the Optimal Pebbling Number in Graphs