Algebraic connectivity and spectral w-variation of unicyclic graphs

Abstract

This article characterizes the unicyclic graphs G with the following property: when a new edge with positive weight w is added between two non-adjacent vertices of G or when the weight of an existing edge is increased by w, exactly two Laplacian eigenvalues of G each increase by w, while all other eigenvalues remain unchanged. Furthermore, we identify the unicyclic graphs for which one of the altered eigenvalues is the algebraic connectivity.