L-Metric Generator in Graphs and Its Application

In this paper, by applying the notion of metric dimension of a graph, we introduce a unique codefor each vertex of a given graph such that the length of codes be as small as possible and each vertexcan be identified by its code. indeed, for an ordered subset s = {v1, . . . , vk} of vertices in a connectedgraph g and a vertex v of g, the l-metric s-representation of a vertex v ∈ v (g) is the vector rgl (v|s) =(ag(v, v1), . . . , ag(v, vk)) , where ag(v, vi) = min{dg(v, vi), l}, i ∈ {1, . . . , k}. s ⊆ v (g) is a l-metricgenerator for g if the vertices of g have pairwise different l-metric s-representations. a l-metric generatorof smallest order is a l-metric basis for g, its order being the l-metric dimension diml(g) of g. we provethat the length of codes obtained from metric dimensions for some graphs is larger than the length ofcodes obtained from our new definition of metric dimensions and the difference can be large unbounded.