Local Formula for Topological Charge Indices and Extremal Results for Trees and Alternating Cyclics

Source abstract

Topological charge indices are distance-based molecular descriptors used in chemical graph theory. This paper gives a local, matrix-free method for computing them in connected weighted molecular graphs, including graphs with multiple bonds. This avoids dense matrix computations and leads to efficient algorithms. As consequences, we obtain simplified formulas for simple graphs and bipartite graphs, recover and extend known formulas for trees, prove sharp extremal theorems for trees, and derive a complete layer-by-layer description for alternating weighted even cycles. In the alternating cycle family, only even distance layers contribute, each nonzero layer is obtained in closed form, and the total charge index admits an explicit formula. These results describe vanishing behavior in weighted cycle families, include the benzene model and higher annulenes as examples, and show that topological charge indices detect bond-order asymmetry invisible to classical distance-count descriptors.