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


Abstract


Let G=(V,E) be a connected graph. The distance between an edge e=xy and a vertex v is defined as \T{d}(e,v)=\T{min}\{\T{d}(x,v),\T{d}(y,v)\}. A nonempty set S \subseteq V(G) is an edge metric generator for G if for any two distinct edges e_1,e_2 \in E(G), there exists a vertex s \in S such that \T{d}(e_1,s) \neq \T{d}(e_2,s). An edge metric generating set with the smallest number of elements is called an edge metric basis of G, and the number of elements in an edge metric basis is called the edge metric dimension of G and it is denoted by \T{edim}(G). In this paper, we study the edge metric dimension of a blow up of a graph G, and also we study the edge metric dimension of the zero divisor graph of the ring of integers modulo n. 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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