A square difference 3-equitable labeling of a graph G with vertex set V is a bijection f from V to {1, 2,.... | V | } jg such that if each edge uv is assigned the label -1 if |[f (u)]2 [f (v)]2 | = -1(mod 4), the label 0 if |[f (u)]2 [f (v)]2 | = 0(mod 4) and the label 1 if |[f (u)]2 [f (v)]| = 1(mod 4), then the number of edges labeled with i and the number of edges labelled with j differ by a...

A square difference 3-equitable labeling of a graph G with vertex set V is a bijection f from V to {1, 2,.... | V | } jg such that if each edge uv is assigned the label -1 if |[f (u)]2 [f (v)]2 | = -1(mod 4), the label 0 if |[f (u)]2 [f (v)]2 | = 0(mod 4) and the label 1 if |[f (u)]2 [f (v)]| = 1(mod 4), then the number of edges labeled with i and the number of edges labelled with j differ by a...

A square difference 3-equitable labeling of a graph G with vertex set V is a bijection f from V to {1, 2,.... | V | } jg such that if each edge uv is assigned the label -1 if |[f (u)]2 [f (v)]2 | = -1(mod 4), the label 0 if |[f (u)]2 [f (v)]2 | = 0(mod 4) and the label 1 if |[f (u)]2 [f (v)]| = 1(mod 4), then the number of edges labeled with i and the number of edges labelled with j differ by a...

A 3-equitable prime cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, ..., |V |} such that if an edge uv is assigned the label 1 if gcd(f(u), f(v)) = 1 and gcd(f(u) + f(v), f(u)−f(v)) = 1, the label 2 if gcd(f(u), f(v)) = 1 and gcd(f(u) + f(v), f(u)− f(v)) = 2 and 0 otherwise, then the number of edges labeled with i and the number of edges labeled with j differ b...

A square difference 3-equitable labeling of a graph G with vertex set V is a bijection f from V to {1, 2,.... | V | } jg such that if each edge uv is assigned the label -1 if |[f (u)]2 [f (v)]2 | = -1(mod 4), the label 0 if |[f (u)]2 [f (v)]2 | = 0(mod 4) and the label 1 if |[f (u)]2 [f (v)]| = 1(mod 4), then the number of edges labeled with i and the number of edges labelled with j differ by a...

Let Z2 = {0, 1} and G = (V ,E) be a graph. A labeling f : V → Z2 induces an edge labeling f* : E →Z2 defined by f*(uv) = f(u).f (v). For i ε Z2 let vf (i) = v(i) = card{v ε V : f(v) = i} and ef (i) = e(i) = {e ε E : f*(e) = i}. A labeling f is said to be Vertex-friendly if | v(0) − v(1) |≤ 1. The vertex balance index set is defined by {| ef (0) − ef (1) | : f is vertex-friendly}. In this paper ...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

A tree is a vertex-edge graph that is connected and contains no cycles. A 4−equitable labeling of a graph is an assignment of labels {0, 1, 2, 3} to the vertices. The edge labels are the absolute difference of the labels of the vertices that they are incident to. The labels must be distributed as evenly as possible amongst the vertices and they must also be distributed as evenly as possible amo...

Let G = (V,E) be a graph with n vertices and f : V (G) → {1, 2, . . . , n} be a bijective function. We define the minimum edge difference as ab(G, f) = min{|f(u)− f(v)| : (u, v) ∈ E}. We say that f is a k-antibandwidth labeling on G if ab(G, f) ≥ k. Let ab(G) = maxf{ab(G, f)}. We therefore investigate the lower bound of ab(G). In other words, to what extent can we maximize the minimum edge diff...