### Isomorphic Circulant Graphs and Applications to Homogeneous Linear Systems

#### Abstract

In this research project we will investigate patterns in numerical data regarding the number of isomorphic circulant graphs for a given order *n*. There is no known formula for this, but computer computations have provided useful data which has not been fully analyzed. These graphs will also be used to model certain circulant linear systems of equations to describe their decomposition into subsystems and, ultimately, provide information about the nature of their solution sets. In order to obtain results, detailed illustrations of the isomorphic circulant graphs will be carefully analyzed as well as their corresponding circulant linear systems of equations. The software system Mathematica will be used to generate larger circulant graphs and certain functions of the program will determine the behaviors of more complicated circulant linear systems of equations. The Online Encyclopedia of Integer Sequences will be accessed for information relevant to pattern recognition in numerical isomorphism class data. Four theorems will be presented pertaining to the behavior of the isomorphic circulant graphs, where each vertex within the graph is connected to one other vertex within the graph. The behaviors of the circulant graphs vary based on the number of vertices or the order of the graph, the distance in between each vertex, whether or not the graph breaks into cycles, and how many lines are connecting the vertices within the cycles, or edges. We anticipate these theorems will help us generate number theoretic formulas for isomorphic circulant graphs, where each vertex within the graph is connected to two or more vertices within the same graph. We will incorporate more advance computer programs to obtain more numerical isomorphism class data which should give us more reliable information regarding the sequences already found.

#### Keywords

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