| viXra.org > Combinatorics and Graph Theory > 2001 |
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| viXra.org > Combinatorics and Graph Theory > 2001 |
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[3] viXra:2001.0462 [pdf] submitted on 2020-01-22 07:38:39
Authors: Theophilus Agama
Comments: 8 Pages.
In this paper we study the distribution of boundary points of expansion. As an application, we say something about the lonely runner problem. We show that given $k$ runners $\mathcal{S}_i$ round a unit circular track with the condition that at some time $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $i=1,2\ldots,k-2$, then at that time we have \begin{align}||\mathcal{S}_{i+1}-\mathcal{S}_i||>\frac{\mathcal{D}(n)\pi}{k-1}\nonumber
\end{align}for all $i=1,\ldots, k-1$ and where $\mathcal{D}(n)>0$ is a constant depending on the degree of a certain polynomial of degree $n$. In particular, we show that given at most eight $\mathcal{S}_i$~($i=1,2,\ldots, 8$) runners running round a unit circular track with distinct constant speed and the additional condition $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $1\leq i\leq 6$ at some time $s>1$, then at that time their mutual distance must satisfy the lower bound\begin{align}||\mathcal{S}_{i}-\mathcal{S}_{i+1}||>\frac{\pi}{7C\sqrt{3}}\nonumber
\end{align}for some constant $C>0$ for all $1\leq i \leq 7$.
Category: Combinatorics and Graph Theory
[2] viXra:2001.0437 [pdf] submitted on 2020-01-21 16:42:34
Authors: N. Durga, S. Satham Hussain, Saeid Jafari, Said Broumi
Comments: 26 Pages.
In this manuscript, the operations on neutrosophic vague graphs are introduced. Moreover, Cartesian product, cross product, lexicographic product, strong product and composition of neutrosophic vague graph are investigated and the proposed concepts are illustrated with examples.
Category: Combinatorics and Graph Theory
[1] viXra:2001.0404 [pdf] submitted on 2020-01-20 18:17:28
Authors: Theophilus Agama
Comments: 6 Pages.
We introduce and study the needle function. We prove that this function is a function modeling $n$-step self avoiding walk. We show that the total length of the $l$-step self-avoiding walk modeled by this function is of the order \begin{align}\ll \frac{n^{\frac{3}{2}}}{2}\bigg(\mathrm{\max}\{\mathrm{sup}(x_j)\}_{1\leq j\leq \frac{l}{2}}+\mathrm{max}\{\mathrm{sup}(a_j)\}_{1\leq j\leq \frac{l}{2}}\bigg).\nonumber
\end{align}
Category: Combinatorics and Graph Theory
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