On randomly k -dimensional graphs

• Published in: Applied mathematics letters Vol. 24; no. 10; pp. 1625 - 1629
• Main Authors: Jannesari, Mohsen, Omoomi, Behnaz
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.10.2011; Elsevier
• Subjects: Basis; Basis number; Combinatorics. Ordered structures; Exact sciences and technology; Graphs; Mathematical analysis; Mathematics; Metric dimension; Numerical analysis; Numerical analysis. Scientific computation; Order, lattices, ordered algebraic structures; Resolving number; Resolving set; Sciences and techniques of general use; Resolving set; Basis number; Basis; Metric dimension; Resolving number; Applied mathematics; Connected graph; Vertex(graph)
• ISSN: 0893-9659; 1873-5452
• DOI: 10.1016/j.aml.2011.03.024

For an ordered set W = { w 1 , w 2 , … , w k } of vertices and a vertex v in a connected graph G , the ordered k -vector r ( v | W ) : = ( d ( v , w 1 ) , d ( v , w 2 ) , … , d ( v , w k ) ) is called the (metric) representation of v with respect to W , where d ( x , y ) is the distance between the vertices x and y . The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W . A resolving set for G with minimum cardinality is called a basis of G and its cardinality is the metric dimension of G . A connected graph G is called a randomly k -dimensional graph if each k -set of vertices of G is a basis of G . In this work, we study randomly k -dimensional graphs and provide some properties of these graphs.