GRAPHS WITH MINIMUM OR MAXIMUM LOCAL METRIC DIMENSION - Anggia
Let W = fw1;w2; :::;wkg be an ordered set consisting of k distinct vertices in a nontrivial connected graph G. The metric representation of a vertex v in G with respect to W is the k-vector r(vjW) = (d(v;w1); d(v;w2); :::; d(v;wk)) where d(v;wi) represents the distance between v and wi for some 1 i k. If for any arbitrary pair of adjacent vertices u and v in G satisfies r(ujW) 6= r(vjW), then W is a local metric set of G. The smallest positive integer k such that G has a local metric k-set is called the local metric dimension of G (lmd(G)). A local metric set of G having cardinality lmd(G) is a local metric basis of G. Moreover, let n be the order of G, l be the number of true twin equivalence classes in G, and d be the diameter of G. Then, G satisfies lmd(G) n????l and lmd(G) n ???? d: In this final project, some nontrivial connected graphs G with lmd(G) = n ???? l or lmd(G) = n ???? d will be presented.