| viXra.org > Combinatorics and Graph Theory > 2012 |
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| viXra.org > Combinatorics and Graph Theory > 2012 |
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[6] viXra:2012.0137 [pdf] submitted on 2020-12-18 20:40:38
Authors: Tai-Choon Yoon, Yina Yoon
Comments: 10 Pages. [Heading "Abstract" added by viXra Admin]
We reviewed the Euler integral for the factorial, Gauss' Pi function, Legendre's gamma function and beta function, and found that gamma function is defective in $\Gamma(0)$ and $\Gamma(-x)$ because they are undefined or indefinable. And we came to a conclusion that the definition of a negative factorial, that covers the domain of the negative space, is needed to the Euler integral for the factorial, as well as the Euler $Y$ function and the Euler $Z$ function, that supersede Legendre's gamma function and beta function.
Category: Combinatorics and Graph Theory
[5] viXra:2012.0136 [pdf] submitted on 2020-12-18 20:43:07
Authors: Tai-Choon Yoon, Yina Yoon
Comments: 4 Pages. [Heading "Abstract" added by viXra Admin]
The Riemann product formula is provided with the function $\zeta(s)$ for all complex numbers from $\int_0^\infty\frac{x^{s-1}}{e^x-1}dx$ by substituting $-x$ only partly for $x$ in the numerator, which is incorrect, because we can find even negative factorials at the same place instead of the so-called trivial zero.
And for Riemann hypothesis, we may derive out $q=\frac{2n\pi}{ln(a)}$ from the complex variable $z=\pm(p+iq)$ of the Riemann zeta function, which is applicable for all positive and negative planes and complex space including $\zeta(\frac{1}{2})$.
Category: Combinatorics and Graph Theory
[4] viXra:2012.0135 [pdf] submitted on 2020-12-18 20:46:20
Authors: Tariq Alraqad, Akbar Ali, Hicham Saber
Comments: 10 Pages.
Let $G$ be a graph containing no component isomorphic to the path graph of order $2$. Denote by $d_w$ the degree of a vertex $w$ in $G$. The augmented Zagreb index ($AZI$) of $G$ is the sum of the quantities $(d_ud_v/(d_u+d_v-2))^3$ over all edges $uv$ of $G$. Denote by $\mathcal{G}(n,\chi)$ the class of all connected graphs of a fixed order $n$ and with a fixed chromatic number $\chi$, where $n\ge5$ and $3\le \chi \le n-1$. The problem of finding graph(s) attaining the maximum $AZI$ in the class $\mathcal{G}(n,\chi)$ has been solved recently in [F. Li, Q. Ye, H. Broersma, R. Ye, MATCH Commun. Math. Comput. Chem. 85 (2021) 257--274] for the case when $n$ is a multiple of $\chi$. The present paper gives the solution of the aforementioned problem not only for the remaining case (that is, when $n$ is not a multiple of $\chi$) but also for the case considered in the aforesaid paper.
Category: Combinatorics and Graph Theory
[3] viXra:2012.0105 [pdf] replaced on 2020-12-18 01:09:28
Authors: Anwesh Bhattacharya
Comments: 10 Pages. Presented at MMLA 2019, PES University, Bangalore.
In this paper, we have derived a formula to find combinatorial sums of the type $\sum_{r=0}^n r^k {n\choose r}$ for $k \in \mathbb{N}$. The formula is conveniently expressed as a linear combination of terms involving the falling factorial. The co-efficients in this linear expression satisfy a recurrence relation, which is identical to that of the Stirling numbers of the first and second kind.
Category: Combinatorics and Graph Theory
[2] viXra:2012.0081 [pdf] submitted on 2020-12-11 08:42:06
Authors: Franz Hermann
Comments: 27 Pages.
It is known that a regular polygon with an even number of sides can be paved with a rhombic mosaic with the rhombus side equal to the polygon side. This paper provides a generalization for constructing a rhombic mosaic for the case of any n-gon. in addition, we consider cases of constructing a rhombic-fractal mosaic of regular polygons. In addition, the hypothesis "about a combined mosaic" is formulated»
Category: Combinatorics and Graph Theory
[1] viXra:2012.0071 [pdf] submitted on 2020-12-10 08:39:05
Authors: Majid Zohrehbandian
Comments: 5 Pages.
Vertex cover problem is a famous combinatorial problem, which its complexity has been heavily studied. It is known that it is hard to approximate to within any constant factor better than 2. In this paper, based on the addition of new constraints to the combination of 3 semidefinite programming (SDP) relaxations, we introduce a new stronger SDP relaxation for vertex cover problem which solve it exactly on general graphs. In this manner and by solving one of the NP-complete problems in polynomial time, we conclude that P=NP.
Category: Combinatorics and Graph Theory
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