General multiplicative Zagreb indices of trees and unicyclic graphs with given matching number

Abstract

The first general multiplicative Zagreb index of a graph G is defined as $P_1^a (G) = \prod _{v \in V(G)} (deg_G (v))^a$ and the second general multiplicative Zagreb index is $P_2^a (G) = \prod _{v \in V(G)} (deg_G (v))^{a \, deg_G (v)}$, where V(G) is the vertex set of G, $deg_{G} (v)$ is the degree of v in G and $a \ne 0$ is a real number. We present lower and upper bounds on the general multiplicative Zagreb indices for trees and unicyclic graphs of given order with a perfect matching. We also obtain lower and upper bounds for trees and unicyclic graphs of given order and matching number. All the trees and unicyclic graphs which achieve the bounds are presented, thus our bounds are sharp. Bounds for the classical multiplicative Zagreb indices are special cases of our theorems and those bounds are new results as well.