For a graph G = (V, E) and a coloring f : V (G) → Z 2 let vf (i) = |f−1(i)|. f is said to be friendly if |vf (1)−vf (0)| ≤ 1. The coloring f : V (G) → Z 2 induces an edge labeling f∗ : E(G) → Z 2 defined by f∗(xy) = f(x) + f(y) ∀xy ∈ E(G), where the summation is done in Z 2. Let ef (i) = |f∗−1(i)|. The friendly index set of the graph G, denoted by FI(G), is defined by FI(G) = {|ef (1)− ef (0)| ...

LetG=(V ,E) be a graph, a vertex labeling f : V → Z2 induces an edge labeling f ∗ : E → Z2 defined by f ∗(xy)=f (x)+f (y) for each xy ∈ E. For each i ∈ Z2, define vf (i)=|f−1(i)| and ef (i)=|f ∗−1(i)|. We call f friendly if |vf (1)− vf (0)| 1. The full friendly index set of G is the set of all possible values of ef (1)− ef (0), where f is friendly. In this note, we study the full friendly index...

For a graph G = (V,E) and a binary labeling (coloring) f : V (G) → Z2, let vf (i) = |f−1(i)|. f is said to be friendly if |vf (1) − vf (0)| ≤ 1. The labeling f : V (G) → Z2 induces an edge labeling f∗ : E(G) → Z2 defined by f∗(xy) = |f(x)− f(y)| ∀xy ∈ E(G). Let ef (i) = |f∗−1(i)|. The friendly index set of the graph G, denoted by FI(G), is defined by FI(G) = {|ef (1)− ef (0)| : f is a friendly ...

Let G = (V,E) be a simple graph. An edge labeling f : E → {0, 1} induces a vertex labeling f : V → Z2 defined by f(v) ≡ ∑ uv∈E f(uv) (mod 2) for each v ∈ V , where Z2 = {0, 1} is the additive group of order 2. For i ∈ {0, 1}, let ef (i) = |f−1(i)| and vf (i) = |(f+)−1(i)|. A labeling f is called edge-friendly if |ef (1) − ef (0)| ≤ 1. If (G) = vf (1) − vf (0) is called the edge-friendly index o...

Let G be a graph with vertex set V(G) and edge set E(G), and let A be an abelian group. A labeling f: V(G) A induces an edge labeling f"': E(G) A defined by f"'(xy) = f(x) + fey), for each edge xy e E(G). For i e A, let vt<i) = card { v e V(G) : f(v) = i} and er(i) = card ( e e E(G) : f"'(e) = i}. Let c(f) = {Iet<i) etG)1 : (i, j) e A x A}. A labeling f of a graph G is said to be A­ friendly if...

Let G = (V, E) be a connected simple graph. A labeling f : V → Z2 induces an edge labeling f∗ : E → Z2 defined by f∗(xy) = f(x) + f(y) for each xy ∈ E. For i ∈ Z2, let vf (i) = |f−1(i)| and ef (i) = |f∗−1(i)|. A labeling f is called friendly if |vf (1)− vf (0)| ≤ 1. The full friendly index set of G consists all possible differences between the number of edges labeled by 1 and the number of edge...

Let G be a graph with vertex set V(G) and edge set E(G), and let A be an abelian group. A labeling f : V(G) ....... A induces a edge labeling r : E{G) ....... A defined by r(xy) = f(x) + f(y) for each xy E E. For each i E A, let vJ(i) card{v E V(G) : f(v) i} and eJ(i) card{e E E(G) : r(e) i}. Let c(J) {leJ(i) eJ(j)1 : = = (i,j) E A x A}. A labeling f of a graph G is said to be A-friendly if IVJ...