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18

May 27, 2021
05/21

May 27, 2021
by
A.S. Muktibodh; S.T. Rathod

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In this paper we define two kinds of pseudo-Smarandache functions. We have investigated more than fifty terms of each pseudo-Smarandache function. We have proved some interesting results and properties of these functions.

In the present paper, we investigate special Smarandache curves with Darboux apparatus with respect to Frenet and Darboux frame of an arbitrary curve on a surface in the three-dimensional Galilean space G3. Furthermore, we give general position vectors of special Smarandache curves of geodesic, asymptotic and curvature line on the surface in G3. As a result of this, we provide some related examples of these curves.

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15

May 27, 2021
05/21

May 27, 2021
by
Pedro Melendez

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A new problem related to the Smarandache Function.

In this paper, we define Smarandache curves of null quaternionic curves in the semi-Euclidean space and obtain that curvatures of null quaternionic curves have some relations for Smarandache curves.

In the light of great importance of curves and their frames in many different branches of science, especially differential geometry as well as geometric properties and the uses in various fields, we are interested here to study a special kind of curves called Smarandache curves in Lorentz 3-space. Then, we present some characterizations for these curves and calculate their Darboux invariants. Moreover, we classify TP, TU, PU and TPU-Smarandache curves of a spacelike curve according to the...

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14

May 27, 2021
05/21

May 27, 2021
by
Ken Tauscher

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Let N be a positif integer. Let η be the function that associates to any non-null integer P the smallest number Q such find the minimwn value of K from which η(R) > N for any R > K.

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13

May 27, 2021
05/21

May 27, 2021
by
Pedro Melendez

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Another problem related to the Smarandache Function.

We introduce special Smarandache curves based on Sabban frame and we investigate geodesic curvatures of Smarandache curves on de Sitter and hyperbolic spaces.The existence of duality between Smarandache curves on de Sitter space and Smarandache curves on hyperbolic space is shown. Furthermore, we give examples of our main results.

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12

May 27, 2021
05/21

May 27, 2021
by
J. Thompson

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This paper proposes a new problem related to the Smarandache function and offer a solution.

In this study, we give special Smarandache curves according to the Sabban frame in hyperbolic space and new Smarandache partners in de Sitter space. The existence of duality between Smarandache curves in hyperbolic and de Sitter space is obtained. We also describe how we can depict picture of Smarandache partners in de Sitter space of a curve in hyperbolic space. Finally, two examples are given to illustrate our main results.

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18

May 27, 2021
05/21

May 27, 2021
by
Thomas Martin

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We propose a problem related to the Smarandache function and offer a solution.

In this study, we introduce the spherical images of some special Smarandache curves according to Frenet frame and Darboux frame in E3. Besides, we give some differential geometric properties of Smarandache curves and their spherical images.

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11

May 27, 2021
05/21

May 27, 2021
by
A. Stuparu; D. W. Sharpe

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We prove that the equation S(x) = p, where p is a given prime number has just D((p-1)!) solutions, all of them in between p and p!.

In this work, we introduce some special Smarandache curves in the Euclidean space. We study Frenet-Serret invariants of a special case. Besides, we illustrate examples of our main results.

12
12

May 27, 2021
05/21

May 27, 2021
by
T. Yau

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We ask to the question: For what triplets Smarandache function verifies the Fibonacci relationship?

In this paper, we introduce some special Smarandache curves according to Bishop frame in Euclidean 3-space E3. Also, we study Frenet-Serret invariants of a special case in E3. Finally, we give an example to illustrate these curves.

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15

May 27, 2021
05/21

May 27, 2021
by
J. Rodriguez

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A problem related to the Smarandache Function.

In this study, we determine TN-Smarandache curves whose position vector is composed by Frenet frame vectors of another regular curve in Minkowski 3-space R. Then, we present some characterisations of Smarandache curves and calculate Frenet invariants of these curves. Moreover, we classify TN; TB; NB and TNB-Smarandache curves of a regular curve parametrized by arc length by presenting a brief table with respect to the causal character. Also, we will give some examples related to results. In...

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18

May 27, 2021
05/21

May 27, 2021
by
Charles Ashbacher

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Being the third in a series on the Smarandache Notions, it is a tribute to the mind of Florentin Smarandache that there seems to be no end to the chain of problems. He is to be commended for contributing so many problems in so many areas. It will be at least decades before most of the problems that he has posed will be resolved. If you found this book interesting, I strongly encourage you to examine the references listed at the end of this book. There is much more there that remains unexplored....

In this paper, we investigate special spacelike Smarandache curves of timelike curves according to Sabban frame in Anti de Sitter 3-Space. Moreover, we give the relationship between the base curve and its Smarandache curve associated with theirs Sabban Frames. However, we obtain some geometric results with respect to special cases of the base curve. Finally, we give some examples of such curves and draw theirs images under stereographic projections from Anti de Sitter 3-space to Minkowski...

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12

May 27, 2021
05/21

May 27, 2021
by
Charles Ashbacher

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This paper deals with palindromic numbers and iterations of the Pseudo-Smarandache Function.

In this study, we introduce new Smarandache curves of a spacelike curve according to the Bishop frame of type-2 in E31. Also, Smarandache breadth curves are defined according to this frame in Minkowski 3-space. A third order vectorial differential equation of position vector of Smarandache breadth curves has been obtained in Minkowski 3-space.

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22

May 27, 2021
05/21

May 27, 2021
by
Charles Ashbacher

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In this paper, some iterations of this function on palindromes that yield palindromes are demonstrated.

In this paper, we investigate Smarandache curves according to type-2 Bishop frame in Euclidean 3- space and we give some differential geometric properties of Smarandache curves. Also, some characterizations of Smarandache breadth curves in Euclidean 3-space are presented. Besides, we illustrate examples of our results.

9
9.0

May 27, 2021
05/21

May 27, 2021
by
Mihaly Bencze

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This paper proposes some open questions for the Smarandache Function.

9
9.0

May 27, 2021
05/21

May 27, 2021
by
E. Radescu; N. Radescu; C. Dumitrescu

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This paper analyzes the Summatory Function associated to the Smarandache Function.

Smarandache Geometries and Curves

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13

May 27, 2021
05/21

May 27, 2021
by
Esra Betul Koc Ozturk; Ufuk Ozturk; Kazim Ilarslan; Emilija Nesovic

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In this paper we define nonnull and null pseudospherical Smarandache curves according to the Sabban frame of a spacelike curve lying on pseudosphere in Minkowski 3-space. We obtain the geodesic curvature and the expressions for the Sabban frame’s vectors of spacelike and timelike pseudospherical Smarandache curves. We also prove that if the pseudospherical null straight lines are the Smarandache curves of a spacelike pseudospherical curve 𝛼, then 𝛼 has constant geodesic curvature....

13
13

May 27, 2021
05/21

May 27, 2021
by
Weiguo Duany; Yanrong Xue

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In this paper, we use the properties and the curve figure of these two functions to study the solvability of an equation.

Smarandache Geometries and Curves

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11

May 27, 2021
05/21

May 27, 2021
by
Esra Betul Koc Ozturk; Ufuk Ozturk; Kazim Ilarslan; Emilija Nesovic

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We define pseudohyperbolical Smarandache curves according to the Sabban frame in Minkowski 3-space.We obtain the geodesic curvatures and the expression for the Sabban frame vectors of special pseudohyperbolic Smarandache curves. Finally, we give some examples of such curves.

9
9.0

May 27, 2021
05/21

May 27, 2021
by
Lu Yaming

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In this paper, we discussed the solutions of the following equation involving the Smarandache function and proved that the equation has infinity positive integer solutions.

In this paper, when the unit Darboux vector of the partner curve of Mannheim curve are taken as the position vectors, the curvature and the torsion of Smarandache curve are calculated. These values are expressed depending upon the Mannheim curve. Besides, we illustrate example of our main results.

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7.0

May 27, 2021
05/21

May 27, 2021
by
Zhang Wenpeng; Xu Zhefeng

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The main purpose of this paper is using the elementary method to study the mean value properties of the Smarandache function, and give an interesting asymptotic formula.

In this paper, let (α, α∗) be Bertrand curve pair, when the unit Darboux vector of the α∗ curve are taken as the position vectors, the curvature and the torsion of Smarandache curve are calculated. These values are expressed depending upon the α curve. Besides, we illustrate example of our main results.

8
8.0

May 27, 2021
05/21

May 27, 2021
by
Albert A. Mullin

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This brief note points out several basic connections between the Smarandache function, fixed-point theory and prime-number theory.

In this paper, a family of ruled surfaces generated by some special curves using a Frenet frame of that curves in Euclidean 3-space is investigated. Some important results are obtained in the case of general helices as well as slant helices. Moreover, as an application, circular general helices, spherical general helices, Salkowski curves and circular slant helices, which illustrate the results, are provided and graphed.

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10.0

May 27, 2021
05/21

May 27, 2021
by
Jinrui Wang

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The main purpose of this paper is using the elementary method to study the estimate problem of S (Fn), and give a sharper lower bound estimate for it.

In this study, we investigated special Smarandache curves belonging to Sabban frame drawn on the surface of the sphere by Darboux vector of Mannheim partner curve. We created Sabban frame belonging to this curve. It were explained Smarandache curves position vector is consisted by Sabban vectors belonging to this curve. Then, we calculated geodesic curvatures of this Smarandache curves. Found results were expressed depending on the Mannheim curve.

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8.0

May 27, 2021
05/21

May 27, 2021
by
Mingdong Xiao

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Let n be any positive integer, Pd(n) denotes the product of all positive divisors of n. The main purpose of this paper is using the elementary and analytic methods to study the mean value properties of a new arithmetical function S (Pd(n)), and give an interesting asymptotic formula for it.

A regular curve in complex space, whose position vector is composed by Cartan frame vectors on another regular curve, is called a isotropic Smarandache curve. In this paper, I examine isotropic Smarandache curve according to Cartan frame in Complex 3-space and give some differential geometric properties of Smarandache curves.

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4.0

May 27, 2021
05/21

May 27, 2021
by
Yani Zheng

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The main purpose of this paper is using the elementary method to study the properties of the Pseudo Smarandache function Z(n), and solve two conjectures posed by Kenichiro Kashihara.

In this paper, Firstly we define NC-Smarandache curve, then we calculate the curvature and torsion of NB and TNB- Smarandache curves together with NC-Smarandache curve. Here T, N and B are Frenet vectors of a curve α and vector C is unit Darboux vector.

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5.0

May 27, 2021
05/21

May 27, 2021
by
J. Sandor

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This is a paper about the Pseudo-Smarandache Function.

In this paper, we study b−Smarandache m1m2 curves of biharmonic new type b−slant helix in the Sol3. We characterize the b−Smarandache m1m2 curves in terms of their Bishop curvatures. Finally, we find out their explicit parametric equations in the Sol3.

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9.0

May 27, 2021
05/21

May 27, 2021
by
Henry Ibstedt

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This study originates from questions posed on alternating iterations involving the pseudo-Smarandache function Z(n) and the Euler function.

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4.0

May 27, 2021
05/21

May 27, 2021
by
Yuanbing Lou

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The main purpose of this paper is using the elementary and analytic methods to study the mean value properties of lnZ(n), and give an interesting asymptotic formula for it.

5
5.0

May 27, 2021
05/21

May 27, 2021
by
Henry Ibstedt

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This study is an extension of work done by Charles Ashbacher. Iteration results have been re-defined in terms of invariants and loops. Further empirical studies and analysis of results have helped throw light on a few intriguing questions.

5
5.0

May 27, 2021
05/21

May 27, 2021
by
Su Gou; Jianghua Li

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The main purpose of this paper is using the elementary method to study the properties of the Pseudo-Smarandache function Z(n), and proved the following two conclusions: The equation Z(n) = Z(n+ 1) has no positive integer solutions; For any given positive integer M, there exists an integer s such that the absolute value of Z(s) − Z(s + 1) is greater than M.

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6.0

May 27, 2021
05/21

May 27, 2021
by
Xuhui Fan

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The main purpose of this paper is using the elementary methods to study the mean value properties of the Pseudo-Smarandache-Squarefree function and Smarandache function, and give two sharper asymptotic formulas for it.

0
0.0

May 27, 2021
05/21

May 27, 2021
by
Yongfeng Zhang

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The purpose of this paper is using the elementary method to study the calculating problem of an infinite series involving the near pseudo Smarandache function K(n), and give an exact calculating formula.

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0.0

May 27, 2021
05/21

May 27, 2021
by
Hai Yang; Ruiqin Fu

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The main purpose of this paper is using the analytic method to study the asymptotic properties of the Near Pseudo Smarandache Function, and give two interesting asymptotic formulae for it.

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6.0

May 27, 2021
05/21

May 27, 2021
by
Lin Cheng

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The main purpose of this paper is using the elementary methods to study the mean value properties of p(n)/Z(n), and give a sharper asymptotic formula for it, where p(n) denotes the smallest prime divisor of n.

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10.0

May 27, 2021
05/21

May 27, 2021
by
J. Sandor

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Recently I. Cojocaru and S. Cojocaru have proved a certian irrationality regarding the Smarandache Function. The author of this note showed that this is a consequence of an old irrationality criteria (which will be used here once again), and proved a result implying the given irrationality.

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4.0

May 27, 2021
05/21

May 27, 2021
by
Chan Shi

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The main purpose of this paper is using the elementary method to study the hybrid mean value properties of the Smarandache kn-digital sequence and Smarandache function, and give an interesting asymptotic formula for it.

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6.0

May 27, 2021
05/21

May 27, 2021
by
Zhongtian Lv

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The main purpose of this paper is using the elementary methods to study a mean value problem involving the F. Smarandache function, and give a sharper asymptotic formula for it.

6
6.0

May 27, 2021
05/21

May 27, 2021
by
Jim Duncan

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The ratio of the number of ones to the number of zeros appears to be approximately 1 for large values of k.

7
7.0

May 27, 2021
05/21

May 27, 2021
by
C. Dumitrescu; C. Rocsoreanu

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The Smarandache function S : N*-N* is defined by the condition that S(n) is the smallest integer m such that m! is divisible by n.

In this paper, we extend the study of Nk-index of a graph for other graph operations. Exact formulas of the Nk-index for corona G ◦ H and neighborhood corona G ⋆ H products of connected graphs G and H are presented. An explicit formula for the splitting graph S(G) of a graph G is computed. Also, the Nk-index formula of the join G + H of two graphs G and H is presented. Finally, we generalize the Nk-index formula of the join for more than two graphs.

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5.0

May 27, 2021
05/21

May 27, 2021
by
Emil Burton

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The study of infinite series involving Smarandache function is one of the most interesting aspects of analysis.

In this paper we explore the concept of strong domination number and investigate strong domination number of some cycle related graphs.

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4.0

May 27, 2021
05/21

May 27, 2021
by
Emil Burton

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The study of infinite series involving Smarandache function is one of the most interesting aspects of analysis.

In this paper, we obtain the domination number, the total domination number and the independent domination number in the neighborhood graph. We also investigate these parameters of domination on the join and the corona of two neighborhood graphs.

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5.0

May 27, 2021
05/21

May 27, 2021
by
A.A.K. Majumdar

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This paper proves the existence of an infinite family of pairs of dissimilar Pythagorean triangles that are pseudo Smarandache related.

In this paper we investigate 4-prime cordial labeling behavior of shadow graph of a path, cycle, star, degree splitting graph of a bistar, jelly fish, splitting graph of a path and star.

In this paper, we consider the notion of the Smarandache curves by considering the asymptotic orthonormal frames of curves lying fully on lightlike cone in Minkowski 3-space R. We give the relationships between Smarandache curves and curves lying on lightlike cone in R.

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9.0

May 27, 2021
05/21

May 27, 2021
by
Marcela Popescu; Paul Popescu; Vasile Seleacu

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In this paper we prove some numerical functions.

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6.0

May 27, 2021
05/21

May 27, 2021
by
Charles Ashbacher

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This paper proves the existence of an infinite family of pairs of dissimilar Pythagorean triangles that are pseudo Smarandacherelated.

In this paper we introduced the new notions semifull signed graph and semifull line (block) signed graph of a signed graph and its properties are obtained. Also, we obtained the structural characterizations of these notions. Further, we presented some switching equivalent characterizations.

5
5.0

May 27, 2021
05/21

May 27, 2021
by
Wei Qin

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The main purpose of this paper is using the elementary method to study the number of the solutions of a given congruent equation and give its all prime number solutions.

A graph with n vertices is said to admit a prime labeling if it’s vertices are labeled with distinct integers 1, 2, · · · , n such that for edge xy , the labels assigned to x and y are relatively prime. The graph that admits a prime labeling is said to be prime. G. Sethuraman has introduced concept of supersubdivision of a graph. In the light of this concept, we have proved that supersubdivision by K2,2 of star, cycle and ladder are prime.

5
5.0

May 27, 2021
05/21

May 27, 2021
by
Jozsef Sandor

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This paper treats some new inequalities and limits for the Smarandache function.

A graph G is a finite non-empty set of objects called vertices together with a set of unordered pairs of distinct vertices of G which is called edges.

6
6.0

May 27, 2021
05/21

May 27, 2021
by
Jozsef Sandor

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The Smarandache function satisfies certain elementary inequalities which have importance in the deduction of properties of this (or related) functions.

In this paper, we introduce the first and second distance eccentricity Zagreb indices of a connected graph G as the sum of the squares of the distance eccentricity degrees of the vertices, and the sum of the products of the distance eccentricity degrees of pairs of adjacent vertices, respectively. Exact values for some families of graphs and graph operations are obtained.

Graphs in this paper are simple and finite. Thus for a graph G, δ(G), ∆(G) and χ(G) denote the minimum degree, maximum degree and chromatic number of G respectively.

7
7.0

May 27, 2021
05/21

May 27, 2021
by
Jozsef Sandor

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These functions and many related others may be studied in the near (or further) future.

9
9.0

May 27, 2021
05/21

May 27, 2021
by
Jozsef Sandor

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Our aim is to show that certain results from a recent paper can be obtained in a simpler way from a generalization.

In this paper, we introduce the minimum equitable dominating Randic energy of a graph and computed the minimum dominating Randic energy of graph. Also, established the upper and lower bounds for the minimum equitable dominating Randic energy of a graph.

6
6.0

May 27, 2021
05/21

May 27, 2021
by
Vasile Seleacu; Narcisa Virlan

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In this paper is studied the limit of a sequence.

Recently, Adiga, et.al. introduced, the minimum covering energy Ec(G) of a graph and S. Burcu Bozkurt, et.al. introduced, Randic Matrix and Randic Energy of a graph. Motivated by these papers, Minimum equitable dominating Randi´c energy of a graph REED(G) of some graphs are worked aut and bounds on REED(G) are obtained.

6
6.0

May 27, 2021
05/21

May 27, 2021
by
Jozsef Sandor

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We obtain a dual of the Smarandache function.

In this paper, we introduce the concept of minimum dominating color energy of a graph, and compute the minimum dominating color energy of few families of graphs. Further, we establish the bounds for minimum dominating color energy.

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7.0

May 27, 2021
05/21

May 27, 2021
by
A.W. Vyawahare; K. M. Purohit

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This new paper defines a new function K(n) where n belings to N, which is a slight modification of Z(n) by adding a smallest natural number k. Hence this function is "Near Pseudo Smarandache Function (NPSF)". Some properties of K(n) are presented here, separately, according to as n is even or odd. A continued fraction consisting NPSF is shown to be convergent. Finally some properties of Kl (n) are also obtained.

Our attempt in this paper is to show that all the linear cyclic snakes, including kC4, are also super vertex mean graphs, even though C4 is not an SVM graph. We also define the term Super Vertex Mean number of graphs.

9
9.0

May 27, 2021
05/21

May 27, 2021
by
Jim Duncan

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Monotonic increasing and monotonic decreasing sequences of S(n) were investigated for (x)= 6.

A complex system S consists m components, maybe inconsistence with m ≥ 2, such as those of biological systems or generally, interaction systems and usually, a system with contradictions, which implies that there are no a mathematical subfield applicable.

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6.0

May 27, 2021
05/21

May 27, 2021
by
Steven R. Finch

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This paper extends earlier work on the average value of the Smarandache function S(n) and is based on a recent asymptotic result.

A new graph characteristic, even modular edge irregularity strength of graphs is introduced. Estimation on this parameter is obtained and the precise values of this parameter are obtained for some families of graphs.

6
6.0

May 27, 2021
05/21

May 27, 2021
by
Yi Yuan; Zhang Wenpeng

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For any positive integer n, let S(n) denotes the Smarandache function, then S(n) is defined the smallest m belonging to N+, where n|m!. In this paper, we study the mean value properties of the additive analogue of S(n), and give an interesting mean value formula for it.

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3.0

May 27, 2021
05/21

May 27, 2021
by
I. Balacenoiu; V. Seleacu

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This paper recapitulates the sientific papers dedicated to the Smarandache function.

An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes of any two color class differ by at most one. In this paper we investigate the equitable chromatic number for the Central graph, Middle graph, Total graph and Line graph of Triple star graph.

This paper introduces equal degree graphs of simple existed graphs. These graphs exhibited some properties which are co-related with the older one. We characterize graphs for which their equal degree graphs are connected, completed, disconnected but not totally disconnected. We also obtain several properties of equal degree graphs and specify which graphs are isomorphic to equal degree graphs and complement of equal degree graphs. Furthermore, the relation between equal degree graphs and degree...

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2.0

May 27, 2021
05/21

May 27, 2021
by
C. Dumitrescu

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This note shows a generalization of the problem 1075.

We discuss cordial labeling of graphs obtained from duplication of certain graph elements in web and armed helm.

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4.0

May 27, 2021
05/21

May 27, 2021
by
R. Muller

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Florentin Smarandache, a mathematician from Eastern Europe, escaped fran his country because the communist authorities had prohibited the publication of his research papers and his participation in international congresses. After two years of waiting in a political refugee camp in Turkey, he emigrated to the United States. As research workers, receiving our co-worker, we decided to publish a selection of his papers.

In this paper we initiate a study on this parameter. In addition, we discuss the related problem of finding the stability of γetc upon edge addition on some classes of graphs.

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May 27, 2021
05/21

May 27, 2021
by
Marius Coman

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It is always difficult to talk about arithmetic, because those who do not know what is about, nor do they understand in few sentences, no matter how inspired these might be, and those who know what is about, do no need to be told what is about. Arithmetic is that branch of mathematics that you keep it in your soul and in your mind, not in your suitcase or laptop. Part One of this book of collected papers aims to show new applications of Smarandache function in the study of some well known...

In this paper, we discuss the adjacency matrices of graceful digraphs such as unidirectional paths,alternating paths,many orientations of directed star and a class of directed bistar. We also discuss the adjacency matrices of unidirectional paths and alternating paths if they are odd digraceful.

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May 27, 2021
05/21

May 27, 2021
by
J. R. Sutton

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This paper shows a way of alculating the Smarandache Function without factorising.

In this paper, we initiate a study of this new parameter and obtain some results concerning this parameter.

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May 27, 2021
05/21

May 27, 2021
by
J. R. Sutton

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The Smarandache function is an integer function, S, of an integer variable, n. S is the smallest integer such that S! is divisible by n.