State transfer on paths with weighted loops

We consider the fidelity of state transfer on an unweighted path \(n \ge 3\) vertices, where a loop weight w has been appended at each end vertices. It is known that if transcendental, then there pretty good from one vertex to other; we prove companion result fact, namely dense subset \([1,\infty )\) such in subset, between vertices impossible. Under mild hypotheses and t, derive upper lower bounds readout time t. Using those bounds, localise times for which close 1. also provide expressions for, on, sensitivity with respect either or w. Throughout, results rely detailed knowledge eigenvalues eigenvectors associated adjacency matrix.