The symmetric division Szeged index: A novel tool for predicting physical and chemical properties of complex networks

This paper introduces a novel graph invariant, the symmetric division Szeged index (<i>SDZ</i>), which generalizes earlier concepts by focusing on vertices positioned closer to an edge's endpoints rather than vertex degrees. It explores several properties and inequalities associated with the <i>SDZ</i>-index, offering examples and results for various graph classes, including bipartite graphs, trees, complete graphs, unicycles, distance-balanced graphs, and triangle-free graphs. Additionally, the <i>SDZ</i>-index is contrasted with other well-known graph indices. The behavior of the <i>SDZ</i>-index under graph operations, such as corona, sum, lexicographic, and Cartesian products, is also examined. The study highlights the index's potential in topological analysis, particularly in revealing statistically significant correlations with the molecular properties of octane isomers. As research progresses, we anticipate further developments and applications that will enhance our comprehension of complex systems and their properties across fields like network science, computer science, and physics.