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

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 ∗ Also at Department of Mathematics, Faculty of Mathematics and Sciences, University of Gadjah Mada, Yogyakarta, Indonesia. D. INDRIATI ET AL. /AUSTRALAS. J. COMBIN. 65 (1) (2016), 14–26 15 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.