A Study on Certain Evaluation Using Distance Concepts in Graphs

Abstract

Different mathematician have rediscovered graph theory many times while dealing newlinewith problems of their areas of work. These problems developed into different areas of graph newlinetheory like graph labelling, graph coloring, metric dimension, domination, graph newlinedecomposition, metro domination etc. newlineLet G and#61501;G(V,E) be a graph. The order of G represented by n and#61501; V and the size of newlineG represented by m and#61501; E . The standard problem concerned the bridges of island named newlineKonigsberg city in Prussia, which is surrounded Pregel river. newlineA connected graph G , the power graph n G is the graph on the vertices of G with newlinethe relation that two vertices are adjacent in n G whenever d(u,v)and#61646;1,2,3,......,n , where newlined(u,v) is the distance between the vertices u and v in G , if d is the diameter of G , then it newlineturns out that d G is a complete graph. newlineWe recall that the minimum cardinality of a minimal subset D of V(G) such that newlineevery vertex in V and#61485;D is adjacent to at least one vertex in D, is called domination number of newlinethe graph G and is denoted by and#61543; (G) . And we recall that the metric dimension of graph G , newlinedenoted by and#61538; (G) , is defined as the cardinality of a minimal subset S of V(G) having the newlineproperty that for each pair of vertices u,v in G there exists w in S such that newlined(u,w) and#61625; d(v,w) . The set S is called the metric basis of G . newlineWe define a vertex dominating set D of G called a metro dominating set newlinewhenever it also serves as a metric basis of G . The minimum cardinality of the metro newlinedominating set is called metro domination number of G , is denoted by (G)