Combinatorics is a field of mathematics that has always fascinated me. Specifically, graph theory, a branch of combinatorics, has always piqued my interest. In general, graph theory deals with the study of mathematical structures, modeled by vertices with edges connecting them. While these graphs can be very simple, they can also get exceedingly complicated in structure; indeed, there are very interesting properties we can say about these graphs. The field is both enormously complex as well as incredibly enlightening … In the summer of ninth grade, I had my first experience with graph theory at a summer math camp called PROMYS. There, I researched the invariant measures of graphs under arbitrary permutations of vertices. An invariant measure is a certain quality of a graph that is preserved by any permutation of the set of vertices. In that project, the question at hand was the following: given a graph G, what methods can be used to determine if the graph has an invariant measure? Furthermore, what constructions of this invariant measure are possible? This topic is of importance in several issues relating to network connectivity. By examining the invariant measures on graphs, one can relate the network connectivity of graphs under, say, arbitrary permutations (or any other measure) and show possible relatedness between structures …

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• Mathematics
• Graph Theory