On the Wiener index of bicyclic graphs and graphs with fixed segment sequence

Abstract

Wiener index is defined as the sum of the distances between all unordered pairs of vertices in a graph. The study of the Wiener index is motivated by its application in chemistry. This thesis focuses on finding extremal bicyclic graphs relative to Wiener index under various conditions such as fixed circumference (length of the longest cycle) or fixed size of the core (maximal subgraph with no degree less than 2). A segment of a graph G is either a path whose end vertices have degree 1 or at least 3 in G and all the internal vertices have degree 2 in G, or a cycle where all the vertices have degree 2 in G except possibly one. The lengths of all the segments of G form it segment sequence. We also discuss extremal graphs with given segment sequence.