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Register a new userGraceful Labeling For bipartite graceful Graphs and related Graphs
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The concept of graceful labels was proposed by Rosa, scholars began to study graceful labels of various graphs and obtained relevant results.Let the graph is a bipartite graceful graph, we have proved some graphs are graceful labeling in this paper.
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In this paper, we investigate odd graceful graph, odd strongly harmonious graph, bipartite graph and their relationship. We proved following results: (1) if G is odd strongly harmonious graph, then G is odd graceful graph ;(2) if G is bipartite odd graceful graph, then G is odd strongly harmonious graph.
A graph $G(V,E)$ of order $|V|=p$ and size $|E|=q$ is called super edge-graceful if there is a bijection $f$ from $E$ to ${0,pm 1,pm 2,...,pm frac{q-1}{2}}$ when $q$ is odd and from $E$ to ${pm 1,pm 2,...,pm frac{q}{2}}$ when $q$ is even such that the induced vertex labeling $f^*$ defined by $f^*(x) = sum_{xyin E(G)}f(xy)$ over all edges $xy$ is a bijection from $V$ to ${0,pm 1,pm 2...,pm frac{p-1}{2}}$ when $p$ is odd and from $V$ to ${pm 1,pm 2,...,pm frac{p}{2}}$ when $p$ is even. indent We prove that all paths $P_n$ except $P_2$ and $P_4$ are super edge-graceful.
Positively-curved, oscillatory universes have recently been shown to have important consequences for the pre-inflationary dynamics of the early universe. In particular, they may allow a self-interacting scalar field to climb up its potential during a very large number of these cycles. The cycles are naturally broken when the potential reaches a critical value and the universe begins to inflate, thereby providing a `graceful entrance to early universe inflation. We study the dynamics of this behaviour within the context of braneworld scenarios which exhibit a bounce from a collapsing phase to an expanding one. The dynamics can be understood by studying a general class of braneworld models that are sourced by a scalar field with a constant potential. Within this context, we determine the conditions a given model must satisfy for a graceful entrance to be possible in principle. We consider the bouncing braneworld model proposed by Shtanov and Sahni and show that it exhibits the features needed to realise a graceful entrance to inflation for a wide region of parameter space.
Two qualitatively different modes of ending superluminal expansion are possible in extended inflation. One mode, different from the one envoked in most extended models to date, easily avoids making big bubbles that distort the cosmic microwave background radiation (CMBR). In this mode, the spectrum of density fluctuations is found to be scale-free, $P(k) propto k^n$, where $n$ might lie anywhere between 0.5 and 1.0 (whereas, previously, it appeared that the range $1.0> n gtsim 0.84$ was disallowed).
Let $P(G,lambda)$ denote the number of proper vertex colorings of $G$ with $lambda$ colors. The chromatic polynomial $P(C_n,lambda)$ for the cycle graph $C_n$ is well-known as $$P(C_n,lambda) = (lambda-1)^n+(-1)^n(lambda-1)$$ for all positive integers $nge 1$. Also its inductive proof is widely well-known by the emph{deletion-contraction recurrence}. In this paper, we give this inductive proof again and three other proofs of this formula of the chromatic polynomial for the cycle graph $C_n$.
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