ON THE EXTREMAL GRAPHS FOR SECOND ZAGREB INDEX WITH FIXED NUMBER OF VERTICES AND CYCLOMATIC NUMBER

The cyclomatic number of a graph G (is denoted by ν) is the minimum number of edges of G whose removal makes G as acyclic. Denote by G_n,ν the collection of all n-vertex connected graphs with cyclomatic number ν. The elements of G_n,ν with maximum second Zagreb (M₂) index (for ν ≤ 4 and ν = k(k−3) 2 +1, where 4 ≤ k ≤ n−2) and with minimum M2 index (for ν ≤ 2) have already been reported in the literature. The main contribution of the present article is the characterization of graphs in the collection Gn,ν with minimum M₂ index for ν ≥ 3 and n ≥ 2(ν−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 ν ≥ 3, where n ≥ 2(ν−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 ν = 5.