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The first multiplicative Zagreb index of a graph $G$ is the product of thesquare of every vertex degree, while the second multiplicative Zagreb index isthe product of the degree of each edge over all edges. In our work, we explorethe multiplicative Zagreb indices of bipartite graphs of order $n$ withdiameter $d$, and sharp upper bounds are obtained for these indices of graphsin $\mathcal{B}(n,d)$, where $\mathcal{B}(n, d)$ is the set of all $n$-vertexbipartite graphs with the diameter $d$. In addition, we explore therelationship between the maximal multiplicative Zagreb indices of graphs\textcolor{blue}{within} $\mathcal{B}(n, d)$. As consequences, those bipartitegraphs with the largest, second-largest and smallest multiplicative Zagrebindices are characterized, and our results extend and enrich some knownconclusions.

Sharp upper bounds for multiplicative Zagreb indices of bipartite graphs with given diameter