Strongly Cospectral Vertices

TL;DR

This paper introduces the concept of strongly cospectral vertices in graphs, exploring their properties, providing constructions, and analyzing their significance in continuous quantum walks.

Contribution

It develops the theory of strongly cospectral vertices, offers methods to construct such graphs, and links vertex similarity to quantum walk curves.

Findings

Strongly cospectral vertices have orthogonal projections differing only by sign in each eigenspace.

Graphs with strongly cospectral vertices can be explicitly constructed.

The similarity of quantum walk curves correlates with vertex properties.

Abstract

Two vertices $a$ and $b$ in a graph $X$ are cospectral if the vertex-deleted subgraphs $X\setminus a$ and $X\setminus b$ have the same characteristic polynomial. In this paper we investigate a strengthening of this relation on vertices, that arises in investigations of continuous quantum walks. Suppose the vectors $e_a$ for $a$ in $V(X)$ are the standard basis for $\mathbb{R}^{V(X)}$. We say that $a$ and $b$ are strongly cospectral if, for each eigenspace $U$ of $A(X)$, the orthogonal projections of $e_a$ and $e_b$ are either equal or differ only in sign. We develop the basic theory of this concept and provide constructions of graphs with pairs of strongly cospectral vertices. Given a continuous quantum walk on on a graph, each vertex determines a curve in complex projective space. We derive results that show tht the closer these curves are, the more "similar" the corresponding vertices are.