Ld(2,1)-labeling on T-graphs

In this paper, we are concerned with the [Formula: see text]-Graphs, which are graphs defined based on the Topological structure of the given set. Precisely, for a given topology [Formula: see text] on a set [Formula: see text], a [Formula: see text]-Graph ‘[Formula: see text]’ is an undirected simple graph with the vertex set [Formula: see text] as [Formula: see text] and the edge set [Formula: see text] as the set of all unordered pairs of nodes [Formula: see text] in [Formula: see text], denoted by [Formula: see text], satisfying either ‘[Formula: see text] and [Formula: see text]’ (or) ‘[Formula: see text] and [Formula: see text]’. The main purpose of this paper is to study the structure of [Formula: see text]-Graphs for various topologies [Formula: see text] on a set [Formula: see text]. Our goals in this paper are threefold. First, to show the [Formula: see text] labeling number [Formula: see text] of any [Formula: see text]-Graph [Formula: see text] exists finitely, if the labeling is [Formula: see text] multiple of non-negative integral values. In addition to show this labeling number [Formula: see text] is not just bounded above but bounded below as well. Second, to measure the bound values in terms of [Formula: see text] multiple of the order of the [Formula: see text]-Graphs and finding a relation between the order of the [Formula: see text]-Graphs and the maximum degree [Formula: see text] of the [Formula: see text]-Graphs. Finally, third is to show that in case of [Formula: see text] [Formula: see text]-graphs on a set with atleast 2 elements, the labeling number is [Formula: see text] and is smaller than that of Griggs and Yeh’s conjecture value [Formula: see text].