Szeged index of a class of unicyclic graphs

Abstract

The Szeged index is a modification of the Wiener index to cyclic molecules.The Szeged index of a connected graph G is defined as S´.G/ D P e2E.G/ n 1 .ejG/n 2 .ejG/,where E.G/ is the edge set of G, and for any e D uv 2 E.G/, n 1 .ejG/ is the number of vertices of G lying closer to vertex u than to vertex v, and n 2 .ejG/ is the number of vertices of G lying closer to vertex v than to vertex u.In this paper, we determine the n-vertex unicyclic graphs whose vertices on the unique cycle have degree at least three with the first, the second and the third smallest as well as largest Szeged indices for n 6, n 7 and n 8, respectively.