Abstract

Isospectral reductions are mathematical techniques used to shrink a matrix, or a graph, into a smaller, dimensionally reduced form while preserving its spectrum. Simple graphs that share isospectral reductions also share quantum walk properties. As such we further develop the inverse process, isospectral unfoldings, to find graphs that share isospectral reductions. We first prove many conditions relating to isospectral reductions and unfoldings. Afterwards, we define two new unfolding processes: general unfolding which unfolds the graph one vertex at a time, and equitable partition unfolding which excels at larger graph unfolding. In particular, through equitable partition unfolding, we discover many graphs that share isospectral reductions with hypercubes onto their equidistance antipodal vertices and therefore have vertices that also have perfect state transfer. Moreover, these graphs are on fewer vertices and edges which may lead to a family of graphs that give better constraints on the relation between the distance between vertices that have perfect state transfer and the number of vertices or edges required.

Degree

College and Department

Computational, Mathematical, and Physical Sciences; Mathematics

Rights

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