This study examined relatively small networks to better understand types of graphs and their properties. Beginning with simple networks of size less than or equal to four vertices, isomorphisms were drawn between like graphs based on their calculated eigenvalues, path length, clustering coefficient, degrees, and components. These properties were then used to identify random, circulent, and small-world networks. This process led to the conclusion that simple graphs may have multiple structures, but are often isomorphic to graphs with the same number of edges and vertices. Additionally, this study found that random graphs have a low average path length and clustering coefficient while circulant graphs have a high path length and clustering coefficient. Between these two extremes emerged a third type of graph: small-world networks, which have low average path length yet a high clustering coefficient. These two properties make small-world networks an ideal network configuration because they are efficient and resistant to failure. Small-world networks, and graphs in general, play a vital role in many other fields, such as neuroscience, sociology, and computer science, which emphasizes the importance of their classification and properties. 

 My project tested two different measures of “small worldness” that I found in literature, effectively trying to reproduce a similar test done in one of the articles I read. I found that we can create a small world graph by taking a lattice graph and choosing a random subset of edges to “rewire” so that the edge starts from the same place but ends somewhere new. After generating many graphs this way, I calculated two measures for “small worldness,” and found that the best parameter of randomness was different between these measurements. This led me to the conclusion that small world graphs must lie within a range, and that different uses of small world graphs may lead us to use different levels of randomness in our rewiring of edges. As a result, I was accepted to present my work at a poster session during the Joint Mathematics Meeting in January of 2023 (poster embeded below).