New bounds on Zagreb connection indices for trees with fixed domination number

A set D of vertices in a graph G is a dominating set of G if every vertex not in D is adjacent to a vertex in D. The domination number, (Formula presented.), is the minimum cardinality among all dominating sets of G. The degree, (Formula presented.), of a vertex v in G is the number of vertices adjacent to v in G, and the connection number, (Formula presented.), of v in G is the number of vertices at distance 2 from v in G. The first Zagreb and modified Zagreb connection indices of G are defined by (Formula presented.) respectively. We obtain new upper bounds for the first and modified Zagreb connection indices of a tree in terms of the its order, the number of leaves and the domination number, and we characterize the extremal trees that achieve equality in the obtained bounds. These results improve results of Raza and Akhter [Chaos, Solitons and Fractals 169 (2023) 113242].