#### Date of Defense

Fall 12-10-1998

#### Department

Mathematics

#### First Advisor

Ping Zhang, Mathematics and Statistics

#### Abstract

For an ordered set *W* = {*w*_{1}, *w*_{2}, ..., *w _{k}*} of vertices and a vertex

*v*in a connected graph

*G*, the (metric) representation of

*v*with respect to

*W*is the

*k*-vector

*r*(

*v*|

*W*) = (

*d*(

*v*,

*w*

_{1}),

*d*(

*v*,

*w*

_{2}),...,

*d*(

*v*,

*w*)), where

_{k}*d*(

*x*,

*y*) represents the distance between the vertices

*x*and

*y*. The set

*W*is a resolving set for

*G*if distinct vertices of

*G*have distinct representations. A new sharp lower bound for the dimension of a graph

*G*in terms of its maximum degree is presented.

A resolving set of minimum cardinality is a basis for* G* and the number of vertices in a basis is its (metric) dimension dim(*G*). A resolving set *S* of* G* is a minimal resolving set if no proper subset of *S* is a resolving set. The maximum cardinality of a minimal resolving set is the upper dimension dim^{+}(*G*). The resolving number res(*G*) of a connected graph *G* is the minimum *k* such that every* k*-set *W* of vertices of *G* is also a resolving set of *G*. Then 1≤dim(*G*)≤dim_(*G*)≤res(*G*)≤*n*-1 for every nontrivial connected graph *G* of order *n*. It is shown that dim^{+}(*G*)=res(*G*)=*n*-1 if and only if *G*=*K _{n}*, while dim

^{+}(

*G*)=res(

*G*)=2 if and only if G is a path of order at least 4 or an odd cycle.

The resolving numbers and upper dimensions of some well known graphs are determined. It is shown that for every pair *a,b* of integers with 2≤*a*≤*b*, there exists a connected graph *G* with dim(*G*)=dim^{+}(*G*)=*a* and res(*G*)=*b*. Also, for every positive integer* N*, there exists a connected graph *G* with res(*G*)-dim^{+}(*G*)≥*N* and dim^{+}(*G*)-dim(*G*)≥*N*.

#### Recommended Citation

Poisson, Christopher, "Resolvability and the Upper Dimension of Graphs" (1998). *Honors Theses*. 292.

https://scholarworks.wmich.edu/honors_theses/292

#### Access Setting

Honors Thesis-Campus Only