Over the last decade, many traditional domains of analysis have been extended to the graph domain. Fields of study that have incorporated the structural information contained within algebraic graphs have seen an explosion of development. In many cases, the graph’s spectrum is a crucial component in these applications, where the eigenvalues and corresponding eigenvectors of the graph’s matrix representation extend the notion of frequency to graph-structured data. Intrinsically, applications that infer or designate an underlying relational structure between graph-structured data may encounter the fundamental problem of shortest paths. Despite the abundance of shortest-path algorithms that currently exist, surprisingly few yet utilize the graph’s spectrum. With the renewed interest in spectral graph theory and the extension of its topics and tools to structured-data analysis in mind, a new method that solves the shortest path problem within the framework of spectral graph theory is presented.
