A Characterization of State Transfer on Double Subdivided Stars

TL;DR

This paper investigates quantum state transfer properties on double subdivided star graphs, revealing no perfect transfer but identifying conditions for pretty good state transfer using spectral analysis.

Contribution

It provides a complete characterization of pretty good state transfer on double subdivided stars, employing Galois group analysis of the characteristic polynomial.

Findings

No perfect state transfer occurs on double subdivided stars.

Conditions for pretty good state transfer are fully characterized.

Spectral analysis using Galois groups is key to the results.

Abstract

A subdivided star $SK_{1,l}$ is obtained by identifying exactly one pendant vertex from $l$ copies of the path $P_3.$ This study is on the existence of quantum state transfer on double subdivided star $T_{l,m}$ which is a pair of subdivided stars $SK_{1,l}$ and $SK_{1,m}$ joined by an edge to the respective coalescence vertices. Using the Galois group of the characteristic polynomial of $T_{l,m},$ we analyze the linear independence of its eigenvalues which uncovers no perfect state transfer in double subdivided stars when considering the adjacency matrix as the Hamiltonian of corresponding quantum system. Then we establish a complete characterization on double subdivided stars exhibiting pretty good state transfer.