On the construction of the reflexive vertex \(k\)-labeling of any graph with pendant vertex (Q2223415)

Summary: A total \(k\)-labeling is a function \(f_e\) from the edge set to first natural number \(k_e\) and a function \(f_v\) from the vertex set to non negative even number up to \(2 k_v\), where \(k=\max \{k_e,2 k_v\}\). A \textit{vertex irregular reflexive \(k\)-labeling} of a simple, undirected, and finite graph \(G\) is total \(k\)-labeling, if for every two different vertices \(x\) and \(x'\) of \(G, wt( x)\neq wt( x')\), where \(wt (x)= f_v (x)+ \Sigma_{xy \in E (G)} f_e(xy)\). The minimum \(k\) for graph \(G\) which has a vertex irregular reflexive \(k\)-labeling is called the reflexive vertex strength of the graph \(G\), denoted by \(\mathrm{rvs}(G)\). In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.