Clusters and various versions of wiener-type invariants

Abstract

The Wiener type invariant W(λ)(G) of a simple connected graph G is defined as the sum of the terms d(u, ν ׀G)λ over all unordered pairs {u,ν} of vertices of G, where d(u, ν ׀G)λ denotes the distance between the vertices u and v in G and λ is an arbitrary real number. The cluster G1{G2} of a graph G1 and a rooted graph G2 is the graph obtained by taking one copy of G1 and |V (G1)| copies of G2, and by identifying the root vertex of the i-th copy of G2 with the i-th vertex of G1, for i = 1; 2; … ; |V (G1)|. In this paper, we study the behavior of three versions of Wiener type invariant under the cluster product. Results are applied to compute several distance-based topological invariants of bristled and bridge graphs by specializing components in clusters.