Tetracyclic graphs with maximum second Zagreb index: A simple approach

In the chemical graph theory, graph invariants are usually referred to as topological indices. The second Zagreb index (denoted by M-2) is one of the most studied topological indices. For n >= 5, let TETn be the collection of all non-isomorphic connected graphs with n vertices and n+3 edges (such graphs are known as tetracyclic graphs). Recently, Habibi et al. [Extremal tetracyclic graphs with respect to the first and second Zagreb indices, Trans. on Combin. 5(4) (2016) 35-55.] characterized the graph having maximum M-2 value among all members of the collection TETn. In this short note, an alternative but relatively simple approach is used for characterizing the aforementioned graph.