The Graceful Tree Conjecture stated by Rosa in the mid 1960s says that every tree can be gracefully labelled. It is one of best known open problems Graph Theory. conjecture has caused a great interest study gracefulness simple graphs and led to many new contributions list graceful graphs. However, it acknowledged not much about structure after 55 years.

We studied the Einstein-Brans-Dicke cosmology in detail. The difference of the evolution of the universe is significant between Einstein-Brans-Dicke cosmology and standard big-bang model during the radiation-dominated era. The power-law evolution of the scale factor is fast enough to solve the cosmological puzzles and slow enough to avoid the graceful exit problem. However, the constraints from...

String cosmology solutions are examined in a generalized phase-space including sources representing arbitrary corrections to lowest order string-dilatongravity effective action. We find a set of necessary conditions for a graceful exit transition from a dilaton-driven inflationary phase to a radiation dominated era. We show that sources allowing such a transition have to violate energy conditio...

If the vertex set V of > =< V,E G can be divided into two un empty sets X and Y , ∅ = = Y V,X Y X ∩ ∪ , but also two nodes of every edge belong to X and Y separately, the G is called bipartite graph. If E ) ,y Y,(x X,y x i i i i ∈ ∈ ∈ ∀ then G is called complete bipartite graph. if n Y m, X = = , the G is marked m,n K . In this paper the graceful labeling, k-graceful labeling, odd graceful labe...

Let G be a (p,q)-graph and k~O. A graph G is said to be k-edge­ graceful if the edges can be labeled by k,k+I, ... ,k+q-I so that the vertex sums are distinct, modulo p. We denote the set of all k such that G is k-edge graceful by egS(G). The set is called the edge-graceful spectrum ofG. In this paper, we are concerned with the problem of exhibiting sets of natural numbers which are the edge-gr...

Further, even though it is intellectually accepted that we have methodological means to combat these factors and provide benefits during the evolutionary process, development managers still choose short term goals over these long term benefits. Given a choice between getting a system out as early as possible and doing a proper design that will evolve gracefully, the first is invariably chosen a...