On friendly index sets of total graphs of trees

摘要

Let G be a graph with vertex set V(G) and edge set E(G), and let A be an abeliart 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 v_f(i) = card{v ∈V(G) : f(v) = i} and e_f(i) = card{e e E(G): f~* (e) = i}. Let c(f) = {|e_f(i)-e_f(j)|: (i, j) e A x A}. A labeling f of a graph G is said to be A-friendly if |v_f(i)-v_f(j)|≤1 for all (i,j) ∈ A × A. If c(f) is a (0, l)-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 {|e_f(0)-e_f(1)|: the vertex labeling f is Z_2-friendly}. In this paper the friendly index sets of the total graphs of some trees are completely determined.