Abstract
Graph theory is one of those subjects that is a vital part of the digital world. It is used to monitor the movement of robots on a network, to debug computer networks, to develop algorithms, and to analyze the structural properties of chemical structures, among other things. It is also useful in airplane scheduling and the study of diffusion mechanisms. The parameters computed in this article are very useful in pattern recognition and image processing. A number d f , w = min d w , t , d w , s is referred as distance between f = t s an edge and w a vertex. d w , f 1 ≠ d w , f 2 implies that two edges f 1 , f 2 ∈ E are resolved by node w ∈ V . A set of nodes A is referred to as an edge metric generator if every two links/edges of Γ are resolved by some nodes of A and least cardinality of such sets is termed as edge metric dimension, e dim Γ for a graph Γ . A set B of some nodes of Γ is a mixed metric generator if any two members of V ∪ E are resolved by some members of B . Such a set B with least cardinality is termed as mixed metric dimension, m dim Γ . In this paper, the metric dimension, edge metric dimension, and mixed metric dimension of dragon graph T n , m , line graph of dragon graph L T n , m , paraline graph of dragon graph L S T n , m , and line graph of line graph of dragon graph L L T n , m have been computed. It is shown that these parameters are constant, and a comparative analysis is also given for the said families of graphs.
Highlights
The characteristics associated to graph distances have piqued the interest of various scholars, and one of them, the metric dimension, has recently been the focus of them. e theory of metric dimension was given by Slater in 1975 [1] and this theory was further elaborated as resolving set of graphs by Harary and Melter in 1976 [2]
In 2016, Imran and Siddiqui [9] computed the metric dimension of some convex polytopes generated by wheel related graphs
We calculate the exact value of vertex, edge, and mixed dimension for dragon graph, Tn,m, line graph of dragon graph, L(Tn,m), paraline graph of dragon graph, L(S(Tn,m)), and line graph of line graph of dragon graph, L(L(Tn,m)). e vertex metric dimension for Tn,m, L(Tn,m), L(S(Tn,m)), and L(L(Tn,m)) is constant and same; dim(Tn,m) dim(L(Tn,m))
Summary
The characteristics associated to graph distances have piqued the interest of various scholars, and one of them, the metric dimension, has recently been the focus of them. e theory of metric dimension was given by Slater in 1975 [1] and this theory was further elaborated as resolving set of graphs by Harary and Melter in 1976 [2]. If all vertices and edges of Γ have different codes of representation with respect to the set X, X is known as a mixed resolving set for graph Γ. Since dim(L(Tn,m)) ≥ 2, it is not a path graph Pn. A a2, bm is resolving set of L(Tn,m), and the representations of all vertices with respect to A are as follows:
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