Strong cospectrality and twin vertices in weighted graphs

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