A Comparative Study of Three Resolving Parameters of Graphs

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 df,w=mindw,t,dw,s is referred as distance between f=ts an edge and w a vertex. dw,f1≠dw,f2 implies that two edges f1,f2∈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, edimΓ 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, mdimΓ. In this paper, the metric dimension, edge metric dimension, and mixed metric dimension of dragon graph Tn,m, line graph of dragon graph LTn,m, paraline graph of dragon graph LSTn,m, and line graph of line graph of dragon graph LLTn,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.