Vertex-decomposable, Cohen--Macaulay and well-covered zero-divisor graphs

Key Points

• The aim is to characterize the vertex-decomposability, Cohen-Macaulayness, and well-coveredness of zero-divisor graphs.
• Defined zero-divisor graphs for finite commutative rings.
• Analyzed graph properties including vertex-decomposability and Cohen-Macaulayness.
• Examined adjacency conditions for vertex connections.
• Characterized conditions under which zero-divisor graphs are vertex-decomposable.
• Established the criteria for Cohen-Macaulayness in zero-divisor graphs.
• Identified conditions for well-coveredness in the context of finite commutative rings.