# On fractional metric dimension of comb product graphs

### Abstract

A vertex $z$ in a connected graph $G$ \textit{resolves} two vertices $u$ and $v$ in $G$ if $d_G(u,z)\neq d_G(v,z)$. \ A set of vertices $R_G\{u,v\}$ is a set of all resolving vertices of $u$ and $v$ in $G$. \ For every two distinct vertices $u$ and $v$ in $G$, a \textit{resolving function} $f$ of $G$ is a real function $f:V(G)\rightarrow[0,1]$ such that $f(R_G\{u,v\})\geq1$. \ The minimum value of $f(V(G))$ from all resolving functions $f$ of $G$ is called the \textit{fractional metric dimension} of $G$. \ In this paper, we consider a graph which is obtained by the comb product between two connected graphs $G$ and $H$, denoted by $G\rhd_o H$. \ For any connected graphs $G$, we determine the fractional metric dimension of $G\rhd_o H$ where $H$ is a connected graph having a stem or a major vertex.### References

S. Arumugam, and V. Mathew, The fractional metric dimension of graphs, Discrete Math., vol. 32, pp. 1584-1590, 2012.

S. Arumugam, V. Mathew, and J. Shen, On fractional metric dimension of graphs, Discrete Math. Algorithms Appl., vol. 5 no. 4, 1350037, 2013.

M. Azari, and A. Iranmanesh, Chemical graphs constructed from rooted product and their Zagreb indices, MATCH Commun. Math. Comput. Chem., vol. 70, pp. 901–919, 2013.

Z. Beerliova, F. Eberhard, T. Erlebach, and L.S. Ram, Network discovery and verification, IEEE J. Sel. Areas Commun., vol. 24, pp. 2168–2181, 2006.

G. Chartrand, L. Eroh, M.A. Johnson, and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., vol. 105, pp. 99–113, 2000.

V. Chvatal, Mastermind, Combinatorica, vol. 3, pp 325–329, 1983.

J. Currie, and O.R. Oellermann, The metric dimension and metric independence of a graph, J. Combin. Math. Combin. Comput., vol. 39, pp. 157–167, 2001.

M. Fehr, S. Gosselin, and O.R. Oellermann, The metric dimension of Cayley graphs, Discrete Math., vol. 306, pp. 31–41, 2006.

M. Feng, B. Lv, and K. Wang, On the fractional metric dimension of graphs, Discrete Appl. Math., vol. 170, pp. 55–63, 2014.

F. Harary, and R.A. Melter, On the metric dimension of a graph, Ars Combin., vol. 2, pp. 191–195, 1976.

S. Khuller, B. Raghavachari, and A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math., vol.70, pp. 217–229, 1996.

D.A. Krismanto, and S.W. Saputro, Fractional metric dimension of tree and unicyclic graph, Procedia Comput. Sci., vol. 74, pp. 47–52, 2015.

A. Sebo, and E. Tannier, On metric generators of graphs, Math. Oper. Res., vol. 29 no.2, pp. 383-393, 2004.

P.J. Slater, Leaves of trees, Congr. Numer., vol. 14, pp. 549–559, 1975.

E. Yi, The fractional metric dimension of permutation graphs, Acta Math. Sinica, vol. 31 no. 3, pp. 367–382, 2015.

*Statistics, Optimization & Information Computing*,

*6*(1), 150-158. https://doi.org/10.19139/soic.v6i1.473

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