3-vertex full balance index set of graphs

Let G be a graph with vertex set V (G) and edge set X(G). Consider the set A = {0, 1, 2}. A labeling f: V (G) → A induces a partial edge labeling f^∗: X(G) → A defined by f^∗(xy) = f(x), if and only if f(x) = f(y), for each edge xy ∈ X(G). For i ∈ A, let v_f (i) = |{v ∈ V (G): f(v) = i}| and e_f^∗ (i) = |{e ∈ X(G): f^∗ (e) = i}|. A labeling f of a graph G is said to be 3-vertex friendly if |v_f (i) − v_f (j)| ≤ 1, for all i ∈ {0, 1, 2}. The 3-vertex full balance index set of a graph G is denoted by F BI_3v (G) and is defined as {e_f^∗(i) − e_f^∗(j), for i, j = 0, 1, 2: f^∗runs over all 3-vertex friendly labeling f of G}. In paper, we study 3-vertex full balance index set and 3-vertex balance index set of some families of graph.