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Given a (molecular) graph, the first multiplicative Zagreb index <inline-formula> <math display="inline"> <semantics> <msub> <mo>&#928;</mo> <mn>1</mn> </msub> </semantics> </math> </inline-formula> is considered to be the product of squares of the degree of its vertices, while the second multiplicative Zagreb index <inline-formula> <math display="inline"> <semantics> <msub> <mo>&#928;</mo> <mn>2</mn> </msub> </semantics> </math> </inline-formula> is expressed as the product of endvertex degree of each edge over all edges. We consider a set of graphs <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">G</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </semantics> </math> </inline-formula> having <i>n</i> vertices and <i>k</i> cut edges, and explore the graphs subject to a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs in <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">G</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </semantics> </math> </inline-formula> are provided. We also provide these graphs with the largest and smallest <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>&#928;</mo> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>&#928;</mo> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> in <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">G</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </semantics> </math> </inline-formula>.