Novel graph distance matrix

Area of Science:

• Graph Theory
• Cheminformatics
• Computational Chemistry

Background:

• Traditional graph representations lack definite geometry.
• Adjacency matrix interpretation in n-dimensional space is underexplored.

Purpose of the Study:

• Introduce a novel distance matrix for graphs.
• Develop new graph invariants for structure-property-activity (SPA) studies.

Main Methods:

• Interpreting graph adjacency matrix columns as points in n-dimensional space.
• Calculating distances using Euclidean distance in n-dimensional space.
• Deriving a natural distance matrix (ND) and a sparse distance-adjacency matrix (DA).

Main Results:

• The natural distance matrix provides definite geometric representation for graphs.
• Novel graph invariants, including a modified connectivity index, were identified.
• The quotient of leading eigenvalues of ND and DA correlates with graph density.

Conclusions:

• The novel distance matrix offers a new perspective on graph structures.
• The developed graph invariants show promise for molecular descriptor applications in SPA studies.
• The approach bridges traditional graph drawing with n-dimensional geometric representation.