Inverse Harary Index Problem for Chain Graphs

A chain graph is a bigraph where the neigh-borhood of vertices in each part forms a chain under set inclusion. Chain graphs have received considerable attention from researchers in the field of spectral graph theory, as they have the maximum spectral radius among all the bigraphs of prescribed size and order. Nevertheless, the areas of some graph parameters remain untouched. The reciprocal Wiener index or the Harary index is one of the distance-based topological indices among several graph parameters designed to capture the different aspects of molecular structure. This article explores the Harary index of chain graphs, giving the bounds and other properties. Further, the Harary index of threshold graphs, a slight structural variant of chain graphs is also discussed. The main focus is on chain graphs with integer-valued Harary index. The article presents a quadratic time algorithm for the inverse Harary index problem for chain graphs and hence contributes significantly to the theory of inverse problems on topological indices.