## Dissertations

#### Title

Measures of Travers Ability in Graphs

6-2007

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

Dr. Ping Zhang

Dr. Gary Chartrand

Dr. Allen Schwenk

Dr. Arthur White

#### Abstract

For a connected graph G of order n ≥ 3 and a cyclic ordering sc : v 1, v2,..., vn, v n+1 = v1 of vertices of G, the number d(sc) is defined by d(sc) = i=1n d(vi, vi +1), where d(vi, vi +1) is the distance between vi and vi+1 in G for 1 ≤ i ≤ n. The Hamiltonian number h(G) and upper Hamiltonian number h +(G) of G are defined as h(G) = min{d(sc)} and h+(G) = max{d(sc)}, respectively, where the minimum and maximum are taken over all cyclic orderings s c of vertices of G. For a connected graph G of order n ≥ 2 and a linear ordering s : v1, v2,..., vn of vertices of G, let d( s) = i=1n-1 d(vi, vi +1). The traceable number t(G) and upper traceable number t+(G) of G are defined as t(G) = min{ d(s)} and t+( G) = max{d(s)}, respectively, where the minimum and maximum are taken over all linear orderings s of vertices of G. We characterize all graphs G for which h+(G) - h(G) = 1 or t+( G) - t(G) = 1. Some relationship between the Hamiltonian number and traceable number of a general graph is also studied.