A vertex magic total (VMT) labeling of a graph G = (V,E) is a bijection from the set of vertices and edges to the set of integers defined by λ : V ∪E → {1, 2, . . . , |V | + |E|} so that for every x ∈ V , w(x) = λ(x)+ ∑ xy∈E λ(xy) = k, for some integer k. A VMT labeling is said to be a super VMT labeling if the vertices are labeled with the smallest possible integers, 1, 2, . . . , |V |. In thi...

An edge-magic total labeling on G is a one-to-one map λ from V (G) ∪ E(G) onto the integers 1, 2, . . . , |V (G) ∪ E(G)| with the property that, given any edge (x, y), λ(x) + λ(x, y) + λ(y) = k for some constant k. The labeling is strong if all the smallest labels are assigned to the vertices. Enomoto et al. proved that a graph admitting a strong labeling can have at most 2|V (G)| − 3 edges. In...

Let E be a finite directed graph with no sources or sinks and write $$X_E$$ for the correspondence. We study $$C^*$$ -algebra $$C^*(E,Z):=\mathcal {T}(X_E,Z)/\mathcal {K}$$ where $$\mathcal {T}(X_E,Z)$$ is generated by weighted shifts on Fock correspondence {F}(X_E)$$ given weight sequence $$\{Z_k\}$$ of operators $$Z_k\in \mathcal {L}(X_{E^{k}})$$ algebra compact If $$Z_k=I$$ every k, $$C^*(E,...

A (p, q)-graph G is called super edge-magic if there exists a bijective function f : V (G) ∪ E(G) → {1, 2, . . . , p + q} such that f(u)+f(v)+f(uv) is a constant for each uv ∈ E(G) and f(V (G)) = {1, 2, . . . , p}. In this paper, we introduce the concept of strong super edgemagic labeling as a particular class of super edge-magic labelings ∗Supported by Slovak VEGA Grant 1/4005/07. †Supported i...

A 1-vertex-magic vertex labeling of a graph G(V,E) with p vertices is a bijection f from the vertex set V (G) to the integers 1, 2, . . . , p with the property that there is a constant k such that at any vertex x the sum ∑ f(x) taken over all neighbors of x is k. In this paper, we study the 1-vertex-magic vertex labelings of two families of disconnected graphs, namely a disjoint union of m copi...

The selective and extensive 13C labeling of mostly hydrophobic amino acid residues in a 25 kDa membrane protein, the colicin Ia channel domain, is reported. The novel 13C labeling approach takes advantage of the amino acid biosynthetic pathways in bacteria and suppresses the synthesis of the amino acid products of the citric acid cycle. The selectivity and extensiveness of labeling significantl...