#### Date of Award

8-1990

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Gary Chartrand

#### Second Advisor

Dr. Alfred Boals

#### Third Advisor

Dr. Naveed Sherwani

#### Fourth Advisor

Dr. Yousef Alavi

#### Abstract

One of the most basic concepts associated with a graph is distance. In this dissertation some new definitions of distance in graphs and digraphs are introduced. One principle goal is to extend certain known results involving the standard distance function on graphs to the field of digraphs with an appropriate concept of distance. Several parameters as well as subgraphs and subdigraphs defined in terms of distance are investigated.

Chapter I gives a brief overview of the history of distance and generalized distance in graphs. By presenting a listing of major results in this area, it provides a background for the chapters to follow.

In Chapter II some results concerning distance in graphs are presented. It is proved that, for a graph G and integers r and d with 1 $\leq$ r $<$ d $\leq$ 2r, there exists a connected graph H of radius r and diameter d such that the center of H is isomorphic to G. A new distance in graphs, called detour distance, is introduced. A generalized Steiner distance in graphs is discussed as well.

In Chapter III maximum distance in digraphs is introduced. It is proved that maximum distance is a metric. The m-radius, m-diameter, m-center, m-periphery and m-median, defined in terms of maximum distance, are studied. In particular, it is proved that every oriented graph is isomorphic to the m-center of some strong oriented graph.

For an oriented graph D, the appendage number of D is defined as the minimum number of vertices required to add to D to produce an oriented graph H such that the m-center of H is isomorphic to D. The main result of Chapter IV is a characterization of oriented acyclic graphs having appendage number 2.

In Chapter V sum distance in digraphs is defined. The s-eccentric set, s-radius, s-diameter, s-center, s-appendage number and s-periphery are investigated. In particular, characterizations of s-eccentric sets and s-peripheries of oriented graphs are presented.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Tian, Songlin, "On Distance in Graphs and Digraphs" (1990). *Dissertations*. 2100.

http://scholarworks.wmich.edu/dissertations/2100