Extending the Unified Mathematical Framework to Support Graph Data Structures

This paper presents a formal extension of the unified query algebra framework to incorporate graph data structures and operations. Building upon the rigorous mathematical foundation established by Jeong (2023), we introduce type-theoretic definitions for graph elements, define traversal and pattern matching operators, establish algebraic properties of graph operations, and extend the category-theoretic foundations to encompass graph structures. We prove that our extension preserves the completeness of the existing Boolean algebra while enabling seamless integration of graph operations with relational, textual, and vector operations. Additionally, we analyze the computational complexity and optimization strategies for cross-paradigm queries involving graphs. This extension provides a comprehensive theoretical foundation for next-generation data systems that must operate across relational, textual, vector, and graph paradigms.