Total Vertex Irregularity Strength of Forests

Original · EN

We investigate a graph parameter called the total vertex irregularity strength (tvs(G)), i.e. the minimal s such that there is a labeling w: E(G)∪ V(G)→ {1,2,..,s} of the edges and vertices of G giving distinct weighted degrees wtG(v):=w(v)+∑ᵥ∈ ₑ ∈ ₑ₍G₎w(e) for every pair of vertices of G. We prove that tvs(F)= (n₁+1)/2 for every forest F with no vertices of degree 2 and no isolated vertices, where n₁ is the number of pendant vertices in F. Stronger results for trees were recently proved by Nurdin et al.