The Wiener index (i.e., the total distance or the transmission number), defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. In this article we summarize some results, conjectures and problems on this molecular descriptor, with emphasis on works we were involved in.

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Mathematical aspects of Wiener index

Neighborhood factors have been linked to HIV risk behaviors, HIV counseling and testing, and HIV medical care. However, the social-psychological mechanisms that connect neighborhood factors to HIV-related behaviors have not been fully determined. In this article we review the research on neighborhood factors and HIV-related behaviors, approaches to measuring neighborhoods, and mechanisms that may help to explain how the physical and social environment within neighborhoods may lead to HIV-related behaviors. We then discuss organizational, geographic, and social network approaches to intervening in neighborhoods to reduce HIV transmission and facilitate HIV medical care with the goal of reducing morbidity and mortality and increasing social and psychological well-being.

Neighborhoods and HIV: a social ecological approach to prevention and care.

In theoretical chemistry, the eccentric connectivity index ξ(G) of a molecular graph G was introduced as ξ ( G ) = ∑ v ∈ V ( G ) d ( v ) ɛ ( v ) where d(v) expresses the degree of vertex v and ɛ(v) is the largest distance between v and any other vertex of G. The corresponding eccentric connectivity polynomial is denoted by ξ ( G , x ) = ∑ v ∈ V ( G ) d ( v ) x ɛ ( v ) . In this paper, we present the exact expressions of eccentric connectivity polynomial for V-phenylenic nanotubes and Zig-Zag polyhex nanotubes.

The eccentric connectivity polynomial of two classes of nanotubes

https://issc.uj.ac.za/sams2009/Abstracts/Morgan.pdf

Simon Mukwembi

Papers

On the eccentric connectivity index of a graph

Note: The edge-Wiener index of a graph

On the degree distance of a graph

Domination with exponential decay

Upper bounds on the average eccentricity