On the Szeged and Wiener complexities in graphs

Abstract

When characterizing networks structurally, the discriminating ability of a topological index is crucial. This relates to investigate its discrimination power (also called uniqueness or degeneracy) that indicates how meaningful the given measure can distinguish nonisomorphic networks. Assume G is a connected graph. The Szeged complexity (or briefly Sz-complexity) of a graph G is the number of different portions in the Szeged index formula. Also, the Wiener complexity (or briefly W-complexity) can be defined similarly. In the current work, we study graphs with small Sz-complexity. We characterize trees with Sz-complexity two and bicyclic graphs with Sz-complexity one. In this way, first we introduce some graphs with Sz-complexity one. For instance, we investigate Θ-graph and two categories of k-cyclic graphs. Besides, we classify bicyclic graphs with Sz-complexity equal to the number of edge-orbits. Finally, we determine W-complexity of these graphs.