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The total vertex irregularity strength of generalized helm graphs and prisms with outer pendant edges

Abstract:

For a simple graph G = (V,E) with the vertex set V and the edge set E, a vertex irregular total k-labeling f: V ∪E → {1, 2,…, k} is a labeling of vertices and edges of G in such a way that for any two different vertices x and x′, their weights wtf (x) = f(x) + ∑xy∈E f(xy) and wtf (x′) = f(x′)+ ∑x′y′∈E f(x′y′) are distinct. A smallest positive integer k for which G admits a vertex irregular total k-labeling is defined as a total vertex irregularity strength of graph G, denoted by tvs(G). In this paper, we determine the exact value of the total vertex irregularity strength for generalized helm graphs and for prisms with outer pendant edges.