Abstract
In this thesis we study and investigate bounds on distance measures in graphs and digraphs. In particular, we relate diameter and size in bipartite graphs and Eulerian bipartite digraphs. We bound proximity and remoteness for triangle-free and C4-free graphs, and finally we bound the difference between remoteness and proximity, diameter and proximity, radius and proximity in triangle-free and C4-free graphs, in terms of other graph parameters. This thesis is made up of four chapters and focuses on finite simple graphs and finite directed graphs. In a connected, finite graph or a strong, finite digraph G of order n, the distance dG(u, v) between two vertices u and v is the length of a shortest u−v path in G. The diameter of G is the largest of the distances between all pairs. In Chapter 1 we introduce and define the fundamentals of graph theory used in this thesis and further review literature on the distance measures investigated in this thesis. In Chapter 2, we give bounds on the number of edges of a bipartite graph G in terms of the order, diameter, edge-connectivity and in terms of the order, diameter, vertex-connectivity...
Ph.D. (Mathematics)