On the maximal augmented Zagreb index of unicyclic graphs with given girth

The augmented Zagreb index ( AZI ) of a graph G is the sum of the weights (d(u)d(v)/(d(u)+d(v)- 2))(3) over all edges u v of G , where d(u) is the degree of the vertex u in G . Let U-n(g) be the set of all unicyclic graphs on n vertices with girth g . In this paper, we prove that if G is a graph with the maximum AZI in U-n(g) , such that n >= 3/2 (g - 2) + 451 and g >= 4, then G is a sunlet graph, i.e. G is a unicyclic graph such that removing all leaves gives a cycle. Furthermore, we have proven that G is always a good sunlet graph, in which all leaves are adjacent to the vertices in the cycle in a regular pattern. Finally we propose a new tool that could help to characterize the graph with the maximum AZI in U-n(g). (c) 2024 Published by Elsevier B.V.