On the vertex irregular reflexive labeling of generalized friendship graph and corona product of graphs

Document Type : Original paper


1 Faculty of Ocean Engineering Technology and Informatics Kuala Ners

2 Universiti Malaysia Terengganu(UMT), Malaysia

3 Universiti Teknologi MARA, Faculty of Computer and Mathematical Sciences, 85100 Segamat, Johor, Malaysia

4 Department of Mathematics GC University, Lahore Pakistan


For a graph $G$, we define a total $k$-labeling $\varphi$ as a combination of an edge labeling $\varphi_e:E(G)\rightarrow \{1,\,2,\,\ldots,\,k_e\}$ and a vertex labeling $\varphi_v:V(G)\rightarrow \{0,\,2,\,\ldots,\,2k_v\}$, where $k=\,\mbox{max}\, \{k_e,2k_v\}$. The total $k$-labeling $\varphi$ is called a vertex irregular reflexive $k$-labeling of $G$ if any pair of vertices $u$, $u'$ have distinct vertex weights $wt_{\varphi}(u)\neq wt_{\varphi}(u')$, where $wt_{\varphi}(u)=\varphi(u)+\sum_{uu'\in E(G)} \varphi(uu')$ for any vertex $u\in V(G)$. The smallest value of $k$ for which such a labeling exists is called the reflexive vertex strength of $G$, denoted by $rvs{(G)}$. In this paper, we present a new lower bound for the reflexive vertex strength of any graph. We investigate the exact values of the reflexive vertex strength of generalized friendship graphs, corona product of two paths, and corona product of a cycle with isolated vertices by referring to the lower bound. This study discovers some interesting open problems that are worth further exploration.


Main Subjects

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