The study defines looped homology for weighted reflexive graphs with parameter r>0 through the categorical graph product. It proves a discrete Mayer-Vietoris exact sequence and Eilenberg-Steenrod axioms, gives examples detecting n-dimensional holes, and shows results can differ from singular homology of the induced topological space.
Key findings
- The theory is non-trivial, detects n-dimensional holes depending on r, admits a discrete Mayer-Vietoris exact sequence and satisfies core axioms. It need not agree with singular homology of the topological space induced by the graph.
Why this matters globally
Topological tools for weighted graphs may inform data analysis, transport, communication and graded-relation networks, provided computable algorithms are developed for large datasets.
Thai researcher contribution
Authors from Chulalongkorn University's mathematics and computer-science community developed the theory and proofs, representing foundational research produced in Thailand.
Limitations to consider
This abstract theory uses constructed examples and provides no algorithm, complexity analysis, software or benchmark against persistent homology and other graph methods. Sensitivity to r and stability under noise remain open.