Strict Lower Bounds On The Multiplicative Zagreb Indices Of Graph Operations

Abstract

The first and second multiplicative Zagreb indices of a simple graph \(G\) are defined as:
\[ \prod_1(G) = \prod_{u \in V(G)} d_G(u)^2
\text{and}
\prod_2(G) = \prod_{uv \in E(G)} d_G(u)d_G(v),\]
where \(d_G(u)\) denotes the degree of the vertex \(u\) of \(G\). In this paper, we establish strict lower bounds on the first and second multiplicative Zagreb indices of various graph operations in terms of the first and second multiplicative Zagreb indices and multiplicative sum Zagreb index of their components.