摘要
A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v E V（G） there is a vertex w E W such that d（u, w） ≠ d（v, w）. A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim（G）. For a vertex u of G and a subset S of V（G）, the distance between u and S is the number minses d（u,s）. A k-partition II = {$1,$2,..., Sk} of V（G） is called a resolving partition if for every two distinct vertices u, v E V（G） there is a set Si in H such that d（u, Si） ≠ d（v, Si）. The minimum k for which there is a resolving k-partition of V（G） is called the partition dimension of G, denoted by pd（G）. The circulant graph is a graph with vertex set Zn, an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i - j （rood n） E C, where C C Zn has the property that C = -C and 0 ￠ C. The circulant graph is denoted by Xn,△ where A = ｜C｜. In this paper, we study the metric dimension of a family of circulant graphs Xn,3 with connection set C = {1, 3, n - 1} and prove that dim（Xn,3） is independent of choice of n by showing that dim（Xn,a） = {3 for all n ≡0 （mod 4）,4 for all n≡2 （mod4）.We also study the partition dimension of a family of circulant graphs Xm,4 with connection set C = {±1, ±2} and prove that pd（Xn,4） is independent of choice of n and show that pd（X5,4） = 5 and pd（Xn,a） ={ 3 for all odd n≥9,4 for all even n≥6 and n=7.
A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v E V（G） there is a vertex w E W such that d（u, w） ≠ d（v, w）. A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim（G）. For a vertex u of G and a subset S of V（G）, the distance between u and S is the number minses d（u,s）. A k-partition II = {$1,$2,..., Sk} of V（G） is called a resolving partition if for every two distinct vertices u, v E V（G） there is a set Si in H such that d（u, Si） ≠ d（v, Si）. The minimum k for which there is a resolving k-partition of V（G） is called the partition dimension of G, denoted by pd（G）. The circulant graph is a graph with vertex set Zn, an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i - j （rood n） E C, where C C Zn has the property that C = -C and 0 ￠ C. The circulant graph is denoted by Xn,△ where A = ｜C｜. In this paper, we study the metric dimension of a family of circulant graphs Xn,3 with connection set C = {1, 3, n - 1} and prove that dim（Xn,3） is independent of choice of n by showing that dim（Xn,a） = {3 for all n ≡0 （mod 4）,4 for all n≡2 （mod4）.We also study the partition dimension of a family of circulant graphs Xm,4 with connection set C = {±1, ±2} and prove that pd（Xn,4） is independent of choice of n and show that pd（X5,4） = 5 and pd（Xn,a） ={ 3 for all odd n≥9,4 for all even n≥6 and n=7.
基金
Supported by the Higher Education Commission of Pakistan (Grant No. 17-5-3(Ps3-257) HEC/Sch/2006)
