Wiener index via wirelength of an embedding

Abstract

Embeddings are often viewed as a high-level representation of systematic methods to simulate an algorithm designed for one kind of parallel machine on a different network structure and/or techniques to distribute data/program variables to achieve optimum use of all available processors. A topological index is a numeric quantity of a molecule that is mathematically derived in an unambiguous way from the structural graph of a molecule. In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. Arguably, the best known of these indices is the Wiener index, defined as the sum of all distances between distinct vertices. In this paper, we have obtained the exact wirelength of embedding Cartesian products of complete graphs into a Cartesian product of paths and cycles, and generalized book. In addition to that, we have found the Wiener index of generalized book and the relation between the Wiener index and wirelength of an embedding, which solves (partially) an open problem proposed in Kumar et al. [K. J. Kumar, S. Klavžar, R. S. Rajan, I. Rajasingh and T. M. Rajalaxmi, An asymptotic relation between the wirelength of an embedding and the Wiener index, submitted to the journal].