Abstract
Let G be a graph containing no component isomorphic to the path graph of order 2. Denote by d(u) the degree of an arbitrary vertex u of G. The augmented Zagreb index (AZI) of G is the sum of the weights (d(u) d(v)/(d(u) + d(v) - 2))(3) over all edges uv of G. In this note, the unique graph with minimal AZI is characterized from the class of all connected tricyclic graphs of order n for every n >= 6, where a connected tricyclic graph of order n is a connected graph of order n and size n + 2 with n >= 4. The obtained result gives a partial solution to a problem posed in the recent paper [W. Lin, D. Dimitrov, R. Skrekovski, Complete characterization of trees with maximal augmented Zagreb index, MATCH Commun. Math. Comput. Chem. 83 (2020) 167-178].