Graphs with Small Distance-Based Complexities

Suppose G is a connected graph. The Wiener complexity (or briefly W-complexity) CW(G) is the number of different contributions to Wiener index in its summation formula. Also, the Szeged complexity (or briefly Sz-complexity) CSz(G) can be defined similarly. The main goal of the current work is to investigate graphs with small complexity. We classify all unicycle graphs, all distance-balanced graphs, and all regular graphs of diameter 2 with Sz-complexity one. In this way, the Sz-complexity and W-complexity of several families of graphs are determined.