A Comparative Study of Three Resolving Parameters of Graphs. (15th December 2021) - Ikhlaq, Hafiz Muhammad; Siddiqui, Hafiz Muhammad Afzal; Imran, Muhammad
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.