On maximum Zagreb connection indices for trees with fixed domination number

The Zagreb connection indices of a graph are notable topological descriptors constructed from the connection number of every vertex (cardinality of set of vertices with distance two from that vertex). In 1972, these indices were presented to determine the total electron energy of the alternate hydrocarbons. The Zagreb connection indices give finer values for the correlation coefficient for the 13 physico-chemical characteristics of the octane isomers compared to basic Zagreb indices. For many years, all these connection indices have been ignored by researchers for further work. Recently, determining the extremal bounds for the topological indices in terms of graph parameters has turned out to be an interesting direction in extremal graph theory, and numerous related results have been acquired in the literature. This article presents sharp bounds on the first, second, and modified Zagreb connection indices of trees with a fixed domination number. These bounds are strict, and the trees which attained these bounds are characterized.