Variable neighborhood search for extremal graphs. 6. Analyzing bounds for the connectivity index

Recently, Araujo and De la Peña gave bounds for the connectivity index of chemical trees as a function of this index for general trees and the ramification index of trees. They also gave bounds for the connectivity index of chemical graphs as a function of this index for maximal subgraphs which are trees and the cyclomatic number of the graphs. The ramification index of a tree is first shown to be equal to the number of pending vertices minus 2. Then, in view of extremal graphs obtained with the system AutoGraphiX, all bounds of Araujo and De la Peña are improved, yielding tight bounds, and in one case corrected. Moreover, chemical trees of a given order and a number of pending vertices with minimum and with maximum connectivity index are characterized.