On vertex irregular total labelings of convex polytope graphs (Q2839666)

Let \(G= G(V,E)\) be a simple, undirected, and finite graph. Here \(V\) denotes the set of vertices with \(p\) elements, and \(E\) the set of edges with \(q\) elements. A labeling of \(G\) refers to any mapping which associates some set of graph elements to a set of numbers (in general positive integers). This labeling is called a vertex-labeling, edge-labeling, or total-labeling accordingly if the domain is \(V\), \(E\), or \(V\cup E\), respectively. The sum of all labels associated with a graph element is called the weight (wt) of that element. A labeling \(\phi:V\cup E\to\{1,2,\dots, k\}\) is called a total \(k\)-labeling. Under a total \(k\)-labeling \(\phi\), the associated weight of a vertex \(x\in V(G)\) is defined as:NEWLINE NEWLINE\[NEWLINE\mathrm{wt}(x)= \phi(x)+ \sum_{y\in N(x)}\phi(xy),NEWLINE\]NEWLINENEWLINE here \(N(x)\) is the set of all neighbors of \(x\). A total \(k\)-labeling \(\phi\) is called a vertex-irregular total labeling of \(G\) if for every two distinct vertices \(x\) and \(y\) of \(G\), we have \(\mathrm{wt}(x)\neq \mathrm{wt}(y)\). The minimum \(k\) for which \(G\) has a vertex-irregular total \(k\)-labeling is defined to be the total vertex irregularity strength of \(G\), denoted by \(\mathrm{tvs}(G)\). NEWLINENEWLINENEWLINE In this paper, the value of the total vertex irregularity strength for some convex polytope graphs is presented.NEWLINENEWLINENEWLINE The following are the main results:NEWLINENEWLINENEWLINE Result 1: Let \(A_n\) be an antiprism graph with \(n\geq 3\). Then the vertex irregularity strength of \(A_n\) is \(\mathrm{tvs}(A_n)= [{2n+4\over 5}]\).NEWLINENEWLINENEWLINE Result 2: The vertex irregularity strength of convex polytopes \(R_n\) with \(n\geq 5\) is \(\mathrm{tvs}(R_n)= [{n+1\over 2}]\).NEWLINENEWLINENEWLINE Result 3: The vertex irregularity strength of a convex polytope \(Q_n\) with \(n\geq 3\) is given by \(\mathrm{tvs}(Q_n)=[{3n+3\over 4}]\).NEWLINENEWLINENEWLINE Result 4: The vertex irregularity strength of a convex polytope \(D_n\) with \(n\geq 3\) is \(\mathrm{tvs}(D_n)= n+1\).