| Summary: | Let <i>R</i> be a commutative ring with identity, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> be the set of zero-divisors of <i>R</i>. The weakly zero-divisor graph of <i>R</i> denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is an undirected (simple) graph with vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>*</mo></msup></mrow></semantics></math></inline-formula>, and two distinct vertices <i>x</i> and <i>y</i> are adjacent, if and only if there exist <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>a</mi><mi>n</mi><mi>n</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>a</mi><mi>n</mi><mi>n</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>s</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. Importantly, it is worth noting that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> contains the zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">W</mi><mo>Γ</mo><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>p</i>, <i>t</i>, and <i>s</i> are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">W</mi><mo>Γ</mo><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>.
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