Abstract
The cyclomatic number of a graph G (is denoted by v) is the minimum number of edges of G whose removal makes G as acyclic. Denote by Gn,v the collection of all n-vertex connected graphs with cyclomatic number v. The elements of Gn,v with maximum second Zagreb (M2) index (for v < 4 and v = k(k???3) 2 + 1, where 4 < k < n??? 2) and with minimum M2 index (for v 2) have already been reported in the literature. The main contribution of the present article is the characterization of graphs in the collection Gn,v with minimum M2 index for v 3 and n > 2(v ??? 1). The obtained extremal graphs, are molecular graphs and thereby, also minimize M2 index among all the connected molecular n-vertex graphs with cyclomatic number v > 3, where n > 2(v ??? 1). For n > 6, the graph having maximum M2 value in the collection Gn,5 has also been characterized and thereby a conjecture posed by Xu et al. [MATCH Commun. Math. Comput. Chem. 72 (2014) 641???654] is confirmed for v = 5.