For $G$ a simple, connected graph, vertex labeling $f:V(G)\to \Z_+$ is called \emph{radio of $G$} if it satisfies $|f(u)-f(v)|\geq\diam(G)+1-d(u,v)$ for all distinct vertices $u,v\in V(G)$. The number the minimal span over radio labelings $G$. If bijective onto $\{1,2,\dots,|V(G)|\}$ exists, graceful} graph. We determine diameter 3 Hamming graphs and show that an infinite subset them graceful.

Programming languages are enormously diverse, both in their essential concepts and in their accidental aspects. This creates a problem when teaching programming. To let students experience the diversity of essential concepts, the students must also be exposed to an overwhelming variety of accidental and irrelevant detail: the accidental differences between the languages are likely to obscure th...

let r be a ring and v be a matrix valuation on r. it is shown that, there exists a correspondence between matrix valuations on r and some special subsets ?(mvpr) of the set of all square matrices over r, analogous to the correspondence between invariant valuation rings and abelian valuation functions on a division ring. furthermore, based on malcolmson’s localization, an alternative proof for t...

This is the fourth part in the series of articles [4], [5], [6] (see also [3]) where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations is an automorphism of the algebra of valuations. Then an integration functional on valuations wi...

In this paper, we prove that the square graph of bistar Bn,n, the splitting graph of Bn,n and the splitting graph of star K1,n are graceful graphs. We also prove that the splitting graph and the shadow graph of bistar Bn,n admit odd graceful labeling.

Many embedded systems have high safety and dependability requirements, which makes ensuring software robustness a top priority in these systems. As embedded computer systems become more complex and incorporate increasing functionality, their software systems become increasingly more difficult to design, build, and maintain. One approach to achieving software robustness is graceful degradation. ...

A rooted tree with diameter D is said to have an even degree sequence if every vertex has even degree except for one root and the leaves, which are in the last level D/2 . The degree sequence is said to be quasi even if every vertex has even degree except for one root, every vertex in level D/2 − 1 and the leaves, which are in the last level D/2 . Hrnčiar and Haviar [P. Hrnčiar, A. Haviar, All ...

We discuss an inflationary scenario based on Lovelock terms. These higher order curvature terms can lead to inflation when there are more than three spatial dimensions. Inflation will end if the extra dimensions are stabilised, so that at most three dimensions are free to expand. This relates graceful exit to the number of large dimensions.