Sums, Products and Ratios along the Edges of a Graph

Comparison of Topological Indices Based on Iterated ‘Sum’ versus ‘Product’ Operations

The Padmakar-Ivan (PI) index is a first-generation topological index (TI) based on sums over all edges between numbers of edges closer to one endpoint and numbers of edges closer to the other endpoint. Edges at equal distances from the two endpoints are ignored. An analogous definition is valid for the Wiener index W, with the difference that sums are replaced by products. A few other TIs are d...

Sums and Products along Sparse Graphs

In their seminal paper from 1983, Erdős and Szemerédi showed that any n distinct integers induce either n distinct sums of pairs or that many distinct products, and conjectured a lower bound of n2−o(1). They further proposed a generalization of this problem, in which the sums and products are taken along the edges of a given graph G on n labeled vertices. They conjectured a version of the sum-p...

Eternal m- Security Subdivision Numbers in Graphs

Let  be a simple graph with vertex set  and edges set . A set  is a dominating set if every vertex in  is adjacent to at least one vertex  in . An eternal 1-secure set of a graph G is defined as a dominating set  such that for any positive integer k and any sequence  of vertices, there exists a sequence of guards   with  and either  or  and  is a dominating set. If we take a guard on every ver...

Computing Vertex PI Index of Tetrathiafulvalene Dendrimers

General formulas are obtained for the vertex Padmakar-Ivan index (PIv) of tetrathiafulvalene (TTF) dendrimer, whereby TTF units we are employed as branching centers. The PIv index is a Wiener-Szeged-like index developed very recently. This topological index is defined as the summation of all sums of nu(e) and nv(e), over all edges of connected graph G.

The Maximum Number of Edges in a Strongly Multiplicative Graph