State transfer in strongly regular graphs with an edge perturbation

Quantum walks, an important tool in quantum computing, have been very successfully investigated using techniques in algebraic graph theory. We are motivated by the study of state transfer in continuous-time quantum walks, which is understood to be a rare and interesting phenomenon. We consider a perturbation on an edge $uv$ of a graph where we add a weight $\beta$ to the edge and a loop of weight $\gamma$ to each of $u$ and $v$. We characterize when for this perturbation results in strongly cospectral vertices $u$ and $v$. Applying this to strongly regular graphs, we give infinite families of strongly regular graphs where some perturbation results in perfect state transfer. Further, we show that, for every strongly regular graph, there is some perturbation which results in pretty good state transfer. We also show for any strongly regular graph $X$ and edge $e \in E(X)$, that $\phi(X\setminus e)$ does not depend on the choice of $e$.