Clusters and Various Versions of Wiener-Type Invariants

Mahdieh Azari, Ali Iranmanesh

The Wiener type invariant $W^{\left(\lambda \right)} (G)$ of a simple connected graph $G$ is defined as the sum of the terms $d(u,v\left|G\right. )^{\lambda }$ over all unordered pairs $\{u,v\}$ of vertices of $G$, where $d(u,v|G)$ denotes the distance between the vertices $u$ and $v$ in $G$ and $\lambda $ is an arbitrary real number. The cluster $G_{1} \{ G_{2} \} $ of a graph $G_{1} $ and a rooted graph $G_{2} $ is the graph obtained by taking one copy of $G_{1} $ and $\left|V(G_{1} )\right|$ copies of $G_{2} $, and by identifying the root vertex of the $i$-th copy of $G_{2} $ with the $i$-th vertex of $G_{1} $, for $i=1,2,…,\left|V(G_{1} )\right|$. 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.