### On the edge metric dimension and Wiener index of the blow up of graphs

#### Abstract

Let be a connected graph. The distance between an edge and a vertex is defined as A nonempty set is an edge metric generator for if for any two distinct edges , there exists a vertex such that . An edge metric generating set with the smallest number of elements is called an edge metric basis of , and the number of elements in an edge metric basis is called the edge metric dimension of and it is denoted by . In this paper, we study the edge metric dimension of a blow up of a graph , and also we study the edge metric dimension of the zero divisor graph of the ring of integers modulo . Moreover, the Wiener index and the hyper-Wiener index of the blow up of certain graphs are computed.

DOI Code:
10.1285/i15900932v40n2p99

Keywords:
Edge metric dimension; Wiener index; Hyper-Wiener index; Blow up of a graph; Zero divisor graph

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