Abstract
The cyclomatic number v of a graph G is the minimum number of those edges of G whose removal makes G as acyclic. The second Zagreb index M-2 for a graph G is the sum of the products of degrees of adjacent vertices of G. For v = ((k-1)(2)) + t with 1 <= t <= k - 1 and 4 <= k <= n - 2, let G* be the graph having maximum M-2 value in the class of all connected graphs with order n and cyclomatic number v. Xu et al. [MATCH Commun. Math. Comput. Chem. 72 (2014) 641-654] posed a conjecture concerning the exact structure of the graph G*. In this note, a partial progress is made on this conjecture by proving that G* has a specific type of subgraph with the size ((k-1)(2)) + t and minimum degree at least one.