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.
![G=(V,E)](http://212.189.136.205/plugins/generic/latexRender/cache/9e9992d6bf50b7580f971487c466a8cb.png)
![e=xy](http://212.189.136.205/plugins/generic/latexRender/cache/cb451ab4bf9f39e541410ce57ee338b2.png)
![v](http://212.189.136.205/plugins/generic/latexRender/cache/9e3669d19b675bd57058fd4664205d2a.png)
![\T{d}(e,v)=\T{min}\{\T{d}(x,v),\T{d}(y,v)\}.](http://212.189.136.205/plugins/generic/latexRender/cache/ab5a7ad268963ecf6993196db0a475c4.png)
![S \subseteq V(G)](http://212.189.136.205/plugins/generic/latexRender/cache/1538506f9b284f3b7c0aca99c45e7ad9.png)
![G](http://212.189.136.205/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![e_1,e_2 \in E(G)](http://212.189.136.205/plugins/generic/latexRender/cache/46fd4a4c72a7a9138085c71c28c3629f.png)
![s \in S](http://212.189.136.205/plugins/generic/latexRender/cache/0718ed5a3e27d89dc5efb8160425a476.png)
![\T{d}(e_1,s) \neq \T{d}(e_2,s)](http://212.189.136.205/plugins/generic/latexRender/cache/534cfe63291e8e92c7fef23952c4ce81.png)
![G](http://212.189.136.205/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![G](http://212.189.136.205/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![\T{edim}(G)](http://212.189.136.205/plugins/generic/latexRender/cache/87780d56bd0e86c659790efad09a1616.png)
![G](http://212.189.136.205/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![n](http://212.189.136.205/plugins/generic/latexRender/cache/7b8b965ad4bca0e41ab51de7b31363a1.png)
DOI Code:
10.1285/i15900932v40n2p99
Keywords:
Edge metric dimension; Wiener index; Hyper-Wiener index; Blow up of a graph; Zero divisor graph
Full Text: PDF