Abstract
The variable sum exdeg index of a graph G is defined as SEIa (G) = Sigma(uv is an element of E(G)) (a(d(u)) + a(d(v))), where d(u) is the degree of a vertex u and a not equal 1 is a positive real number. In [11, maximal trees, unicyclic and bicyclic graphs (i.e., graphs with cyclomatic number 0,1 and 2) and minimal trees and unicyclic graphs (i.e., graphs with cyclomatic number 0 and 1) with respect to variable sum exdeg index for a > 1 were determined Here, we extend those results in two directions. Firstly, for a > 1, we characterize the extremal graphs with a cyclomatic number k <= n - 2, where n is the order of G. Secondly, for 0 < a < 1/e(2) approximate to 0.135335, we characterize the extremal graphs with k <= n - 2, and for 0 < a < 1/3, we characterize the trees, unicyclic, bicyclic, tricyclic and tetracyclic graphs having maximal SEIa, value.