Strong Cospectrality and Twin Vertices in Weighted Graphs

TL;DR

This paper investigates the spectral properties of weighted graphs with twin vertices, extending the concept of strong cospectrality to Hermitian matrices and identifying conditions for such properties in graph joins, with implications for quantum state transfer.

Contribution

It generalizes strong cospectrality to Hermitian matrices and characterizes when twin vertices in weighted graphs exhibit this property, including in graph joins.

Findings

Necessary and sufficient conditions for strong cospectrality in weighted graphs.

Extension of equitable partitions to weighted graphs.

Identification of graph joins exhibiting strong cospectrality.

Abstract

We explore algebraic and spectral properties of weighted graphs containing twin vertices that are useful in quantum state transfer. We extend the notion of adjacency strong cospectrality to arbitrary Hermitian matrices, with focus on the generalized adjacency matrix and the generalized normalized adjacency matrix. We then determine necessary and sufficient conditions such that a pair of twin vertices in a weighted graph exhibits strong cospectrality with respect to the above-mentioned matrices. We also generalize known results about equitable and almost equitable partitions, and use these to determine which joins of the form $X\vee H$, where $X$ is either the complete or empty graph, exhibit strong cospectrality.