A Novel Version of the Edge-Szeged Index

A novel version of the edge-Szeged index is proposed in parallel to the revised (vertex) Szeged index, and some properties, especially lower and upper bounds are established for this molecular descriptor. (doi: 10.5562/cca1889)


INTRODUCTION
The Szeged index, denoted by Sz, has been introduced by Gutman 1 in 1994 whilst he was visiting the Attila József University in Szeged, attractive historical city in south-east Hungary, hence the name of this molecular descriptor.Since the Szeged index has immediately been recognized as interesting and useful molecular descriptor, many contributions appeared to report its mathematical properties (e.g., References 2-9) and various applications to modeling the physicochemical properties and physiological activites of organic compounds.Khadikar et al. 10 succinctly summarized the applications of the Szeged index in the quantitative structure-property-activity-toxicity modeling.In that paper the authors also stated that the total number of publications on the Szeged index and its variants up to the time of preparation of the paper is about 120 and among them more than 60 are reporting various applications.All these papers are listed in their review. 10ince the article by Khadikar et al. 10 appeared in 2005, many more contributions appeared, e.g., References  11-15.The study of the Szeged index and its variants, such as the hyper-Szeged index, PI index, Harary-Szeged index, edge-Szeged index, etc., in the structure-property-activity modeling was summarized besides in Reference 10 also in two comprehensive handbooks on molecular descriptors. 16,17All these warrant further work on the Szeged index and its variants.In this report, we present a novel version of the edge-Szeged index.

PRELIMINARIES
Let G be a connected graph with vertex set ( ) V G and edge set ( ).
E G For , ( ), G n u v be the number of ver- tices closer to vertex u than vertex v and ( , ) The Szeged index of a (molecular

G m v u e E G d e v G d e u G   
The edge-Szeged index of G is defined as 24   ( ) ( , ) ( , ).

G m uv e E G d e u G d e v G   
Motivated by the reports on the revision of the Szeged index, 20,22 below we propose the revised edge-Szeged index of , G defined as

REVISED EDGE-SZEGED INDEX OF CONNECTED GRAPHS
Let n S and n P be respectively the star and the path on n vertices, and n K the complete graph on n vertices.Note also the molecular graphs are necessarily connected graphs. 18oposition 1.Let G be a connected graph with m edges.Then

G n v u the
number of vertices closer to vertex v than vertex , for this index have been established, e.g., References 20-23.For ( ), e uv E G   ( ), w V G  the distance between e and w in G is defined as ( , | ) d e w G  min{ ( , | ), ( , | )}.d u w G d v w G Let ( , ) G m u v be the number of edges closer to vertex u than vertex v and ( , ) G m v u the number of edges closer to vertex v than vertex , u i.e., Thus the study of the revised edge-Szeged index, like the edge-Szeged index and the Szeged index, for trees is equivalent to the study of the Wiener index.It is interesting to see what happens for graphs with cycles.The startpoint is naturally unicyclic graphs.withequality if and only if G is the cycle .
n G K  REVISED EDGE-SZEGED INDEX OF TREES Proposition 2. Let T be an n-vertex tree.Then