On total vertex-irregular labellings for several types of trees

Abstract

For a simple graph G with the vertex set V(G) and the edge set E(G), a labelling A: V(G)UE(G) → {1,2,.... k} is called a vertex-irregular total fc-labelling of λ if for any two different vertices x and y in V(G), we have wt(x) wt(y) where xvt(x) = X(x) + ∑(xz). The tctal vertex-irregular strength, denoted by tvs(G), is the smallest positive integer k for which G has a vertex-irregular total k—labelling. In this paper, we determine the total vertex-irregular strength for various types of trees, namely complete k—ary trees, a subdivision of stars, and a subdivision of particular type of caterpillars. © 2020 Utilitas Mathematica Publishing Inc.. All rights reserved.