A Differential Approach for Bounding the Index of Graphs under Perturbations

This paper presents bounds for the variation of the spectral radius λ(G) of a graph G after some perturbations or local vertex/edge modifications of G. The perturbations considered here are the connection of a new vertex with, say, g vertices of G, the addition of a pendant edge (the previous case with g = 1) and the addition of an edge. The method proposed here is based on continuous perturbations and the study of their differential inequalities associated. Within rather economical information (namely, the degrees of the vertices involved in the perturbation), the best possible inequalities are obtained. Besides, the cases when equalities are attained are characterized. The asymptotic behaviors of the obtained bounds are also discussed. For instance, if G is a connected graph and Gu denotes the graph obtained from G by adding a pendant edge at vertex u with degree δu, then, λ(Gu) ≤ λ(G) + δu λ3(G) + o ( 1 λ3(G) ) .