On extremal multiplicative Zagreb indices of trees with given domination number

Abstract

For a graph G, the first multiplicative Zagreb index 1 is equal to the product of squares of the vertex degrees, and the second multiplicative Zagreb index 2 is equal to the product of the products of degrees of pairs of adjacent vertices. The (mutiplicative) Zagreb indices have been the focus of considerable research in computational chemistry dating back to Gutman and Trinajstić in 1972. In this paper, we explore the mutiplicative Zagreb indices in terms of arbitrary domination number. The sharp upper and lower bounds of 1 (G) and 2 (G) are given. In addition, the corresponding extreme graphs are charaterized.By the recursive process, we see thatRecently, multiplicative Zagreb indices attracted extensive attention in physics, chemistry, graph theory, etc. Xu and Hua [18] proposed a unified approach to characterize extremal (maximal and minimal) trees, unicyclic graphs and bicyclic graphs with respect to multiplicative Zagreb indices, respectively. Iranmanesh et al. [19] investigated these indices the first and the second multiplicative Zagreb indices for a class of dendrimers. Liu and Zhang [14] introduced several sharp upper bounds for π 1 -index and π 2 -index in terms of graph parameters including the order, size and radius [20]. Wang and Wei [21] studied these indices in k-trees and extremal k-trees were characterized. Ramin Kazemi [24] obtained the bounds for the moments and the probability generating function of these indices in a randomly chosen molecular graph with tree structure of order n. Bojana Borovićanin et al. [25] presented upper bounds on Zagreb indices of trees in terms of domination number. Also, a lower bound for the first Zagreb index of trees with a given domination number is determined and the extremal trees are characterized as well. Motivated by the above results, in this paper we further investigate the multiplicative Zagreb indices of trees in terms of domination number. This enriches and extends some earlier results obtained by Bojana Borovićanin et al. [25].

Hamilton-connectivity of Interconnection Networks Modeled by a Product of Graphs

“…A perfect matching between two disjoint graphs G 1 , G 2 with the same order n is a matching consisting of n edges such that each of them has one end vertex in G 1 and the other one in G 2 . See [7][8][9][10][11][12][13][14].…”

The Extremal Graphs of Some Topological Indices with Given Vertex k-Partiteness

“…They are widely used in chemistry to study the heat information of heptanes and octanes. For some recent results on the fourth Zagreb indices, one can see [12][13][14][15][16][17].…”

Quasi-Tree Graphs With Extremal General Multiplicative Zagreb Indices

“…Many investigations of mathematical properties on the multiplicative Zagreb indices have been obtained, recent results see [10,[19][20][21]23] and the references cited therein.…”