On friendly index sets of total graphs of trees
Abstract
Let G be a graph with vertex set V(G) and edge set E(G), and let A be an abelian group. A vertex labeling f : V(G) → A induces an edge labeling f* : E(G) → A defined by f* (xy) = f(x) + f(y), for each edge xy ∈ E(G). For i ∈ A, let Vf(i) = card{v ∈ V(G) : f(v) = i} and ef(i) = card{e ∈ E(G) : f*(e) = i}. Let c(f) = {|ef(i) - ef(j)| : (i, j) ∈ A × A}. A labeling f of a graph G is said to be A-friendly if |vf(i) - Vf(j)| ≤1 for all (i, j) ∈ A × A. If c(f) is a (0, 1)-matrix for an A-friendly labeling f, then f is said to be A-cordial. When A = Z2, the friendly index set of the graph G, FI(G), is defined as {|ef(0) - ef(1)| : the vertex labeling f is Z2-friendly}. In this paper the friendly index sets of the total graphs of some trees are completely determined.
