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...

Given a graph G = (V, E) of order n and a finite abelian group H = (H, +) of order n, a bijection f of V onto H is called a vertex H-labeling of G. Let g(e) ≡ (f(u) + f(v)) mod H for each edge e = {u, v} in E induce an edge H-labeling of G. Then, the sum Hvalf (G) ≡ ∑ e∈E g(e) mod H is called the H-value of G relative to f and the set HvalS(G) of all H-values of G over all possible vertex H-lab...

For any non negative real values h and k, an L(h, k)-labeling of a graph G = (V, E) is a function L : V → R such that |L(u) − L(v)| ≥ h if (u, v) ∈ E and |L(u)−L(v)| ≥ k if there exists w ∈ V such that (u, w) ∈ E and (w, v) ∈ E. The span of an L(h, k)-labeling is the difference between the largest and the smallest value of L. We denote by λh,k(G) the smallest real λ such that graph G has an L(h...

Tree is one of the most studied and practically useful classes of graphs and is the attention of a great number of studies. There is absence of generalized results for tree as a class and even for one kind of labeling as whole. Only specialized results exist limited to specific types of trees only. A number of conjectures stand being unsolved. Graham and Sloane (1980) conjectured trees to be Ha...

We study edge-sum distinguishing labeling, a type of labeling recently introduced by Z. Tuza (2017) in context games. An ESD an $n$-vertex graph $G$ is injective mapping integers $1$ to $l$ its vertices such that for every edge, the sum on endpoints unique. If $ l$ equals $n$, we speak about canonical labeling. focus primarily structural properties this and show several classes graphs if they h...