Summary: If a graph is connected then the largest eigenvalue (i.e., index) generally changes (decreases or increases) if some local modifications are performed. In this paper two types of modifications are considered: (i) for a fixed vertex, \(t\) edges incident with it are deleted, while \(s\) new edges incident with it are inserted; (ii) for two non-adjacent vertices, \(t\) edges incident with one vertex are deleted, while \(s\) new edges incident with the other vertex are inserted. Within each case, we provide lower and upper bounds for the indices of the modified graphs, and then give some sufficient conditions for the index to decrease or increase when a graph is modified as above.