Syed Ahtsham Ul Haq Bokhary

On the metric dimension of circulant graphs

Abstract Let G = ( V , E ) be a connected graph and d ( x , y ) be the distance between the vertices x and y in V ( G ) . A subset of vertices W = { w 1 , w 2 , … , w k } is called a resolving set or locating set for G if for every two distinct vertices x , y ∈ V ( G ) , there is a vertex w i ∈ W such that d ( x , w i ) ≠ d ( y , w i ) for i = 1 , 2 , … , k . A resolving set containing the minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension, denoted by d i m ( G ) . Let F be a family of connected graphs G n : F = ( G n ) n ≥ 1 depending on n as follows: the order | V ( G ) | = φ ( n ) and lim n → ∞ φ ( n ) = ∞ . If there exists a constant C > 0 such that d i m ( G n ) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension. The metric dimension of a class of circulant graphs C n ( 1 , 2 ) has been determined by Javaid and Rahim (2008) [13] . In this paper, we extend this study to an infinite class of circulant graphs C n ( 1 , 2 , 3 ) . We prove that the circulant graphs C n ( 1 , 2 , 3 ) have metric dimension equal to 4 for n ≡ 2 , 3 , 4 , 5 ( mod 6 ) . For n ≡ 0 ( mod 6 ) only 5 vertices appropriately chosen suffice to resolve all the vertices of C n ( 1 , 2 , 3 ) , thus implying that d i m ( C n ( 1 , 2 , 3 ) ) ≤ 5 except n ≡ 1 ( mod 6 ) when d i m ( C n ( 1 , 2 , 3 ) ) ≤ 6 .

ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

Abstract In this paper, we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks J n , a class of cubic convex polytopes considering the open problem raised in [M. Imran et al., families of plane graphs with constant metric dimension, Utilitas Math., in press] and finally Harary graphs H 5, n by partially answering to an open problem proposed in [I. Javaid et al., Families of regular graphs with constant metric dimension, Utilitas Math., 2012, 88: 43–57]. We prove that these classes of regular graphs have constant metric dimension. It is natural to ask for the characterization of regular graphs with constant metric dimension.