Spectral w-variation of trees in two places

This article introduces the concept of spectral w-variation of weighted graphs. A weighted graph G is said to have spectral w-variation in two places if adding an edge of positive weight w between two nonadjacent vertices of G, or increasing the weight of an existing edge by w, results in an increase of two Laplacian eigenvalues of G equally by w while keeping the other eigenvalues unchanged. This article characterizes the weighted graphs that have spectral w-variation in two places. It is proved that spectral w-variation in two places does not occur by increasing the weight of an existing edge in any weighted tree. Mainly, the article determines the weighted trees that have spectral w-variation in two places. As an application, we supply constructions of few classes of weighted trees with weights from the interval $(0,1)$ in which spectral w-variation occurs in two places.