## Honors Theses

Fall 12-10-1998

#### Department

Mathematics

Ping Zhang, Mathematics and Statistics

#### Abstract

For an ordered set W = {w1, w2, ..., wk} 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,w1), d(v,w2),...,d(v,wk)), where 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=Kn, 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≤ab, 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.

#### Access Setting

Honors Thesis-Campus Only

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