A (p, q)-graph G = (V, E) is said to be magic if there exists a bijection f : V ∪ E → {1, 2, 3, . . . , p + q} such that for all edges uv of G, f(u) + f(v) + f(uv) is a constant. The minimum of all constants say, m(G), where the minimum is taken over all such bijections of a magic graph G, is called the magic strength of G. In this paper we define the maximum of all constants say, M(G), analogo...

a graph of size n is said to be graceful when is possible toassign distinct integers from {0, 1, . . . , n} to its verticesand {|f(u)−f(v)| : uv ∈ e(g)} consists of n integers. inthis paper we present broader families of graceful graphs; these families are obtained via three different operations: the third power of a caterpillar, the symmetric product of g and k2 , and the disjoint :union: of g...

It is well known that an (in general, non-commutative) set of non-Hermitian operators $\Lambda_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class quantum models in which these plus the underlying physical Hilbert-space metric $\Theta$ are all represented terms auxiliary operator $(N+1)-$plet $Z_k$, $k=0,1,\ldots,N$. Our formalism degenerates to ${\...

Objective(s): Normoxic MAGIC-f polymer gels are established dosimeters used for three dimensional dose quantifications in radiotherapy. Nanoparticles with high atomic number such as gold are novel radiosensitizers used to enhance doses delivered to tumors. The aim of this study was to investigate the effect of gold nanoparticles (GNPs) in enhancing percentage depth doses (PDDs) within the MAGIC...

Introduction: Normoxic MAGIC-f polymer gels are established dosimeters used for three dimensional dos uantifications in radiotherapy. The high atomic number nanoparticles such as gold are nov adiosensitizers used to enhance doses delivered to tumors. The aim of this study was t vestigate the effect of gold nanoparticles (GNPs) in enhancing percentage depth doses (PDD ithin the...