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.

A total labelling of a graph with v vertices and e edges is a one-to-one map taking the vertices and edges onto the set {1, 2, 3, . . . , v + e}. A labelling can be used to define a weight for each vertex and edge. For a vertex the weight is the sum of the label of the vertex and the labels of the incident edges. For an edge {x, y} the weight is the sum of the label of the edge and the labels o...

An $L(h_1, h_2, \ldots, h_l)$-labelling of a graph $G$ is mapping $\phi: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that for $1\le i\le l$ and each pair vertices $u, v$ at distance $i$, we have $|\phi(u) - \phi(v)| \geq h_i$. The span $\phi$ the difference between largest smallest labels assigned to by $\phi$, $\lambda_{h_1, h_l}(G)$ defined as minimum over all h_l)$-labellings $G$. In this pap...

Graph labellings form an important graph theory model for the channel assignment problem. An optimum labelling usually depends on one or more parameters that ensure minimum separations between frequencies assigned to nearby transmitters. The study of spans and of the structure of optimum labellings as functions of such parameters has attracted substantial attention from researchers, leading to ...

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. ...

We establish new lower and upper bounds for the real number graph labelling problem. As an application, we completely determine the optimum spans of L(p, q)-labellings of the infinite triangular plane lattice (solving an open problem of Griggs).

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.

In this Letter, we describe how a spectrum of entropic perturbations generated during period slow contraction can source nearly scale-invariant curvature on length scales larger than the Hubble radius transition from to classical non-singular bounce (the `graceful exit' phase). The sourcing occurs naturally through higher-order scalar field kinetic terms common (non-singular) mechanisms. We pre...