scholarly journals Coprime Irregular graphs

2019 ◽  
Vol 9 (1S4) ◽  
pp. 1093-1096
Keyword(s):  
Positive Integer ◽  
Edge Weighting ◽  
Irregular Graphs

An l-edge-weighting of a graph G is a map : E(G) {1, 2, 3, … l}, where l is a positive integer. For a vertex v V(G), the weight (v) is the sum of edge-weights appearing on the edges incident at v under the edge-weighting . An l-edge-weighting of G is coprime irregular edge-weighting of G if gcd( (u), (v)) = 1 for every pair of adjacent vertices u and v in G. A graph G is coprime irregular if G admits a coprime irregular edge-weighting. In this paper, we discuss this new irregular edge weighting of graphs.

scholarly journals On Equitable Irregular Graphs

2020 ◽  
Vol 8 (4S4) ◽  
pp. 122-124
Keyword(s):  
Positive Integer ◽  
Edge Weighting ◽  
Irregular Graphs

An k−edge-weighting of a graph G = (V,E) is a map 𝝋: 𝑬(𝑮) → {𝟏,𝟐,𝟑, . . . 𝒌}, }where 𝒌 ≥ 𝟏 is an integer. Denote 𝑺𝝋(𝒗) is the sum of edge-weights appearing on the edges incident at the vertex v under𝝋 . An k-edge -weighting of G is equitable irregular if |𝑺𝝋(𝒖) − 𝑺𝝋(𝒗)| ≤ 𝟏, for every pair of adjacent vertices u and v in G. The equitable irregular strength 𝑺𝒆 (𝑮) of an equitable irregular graph G is the smallest positive integer k such that there is a k-edge weighting of G. In this paper, we discuss the equitable irregular edge-weighting for some classes of graphs

scholarly journals Special Classes of Coprime Irregular Graphs

2020 ◽  
pp. 269-271
Keyword(s):  
Positive Integer ◽  
Edge Weighting ◽  
Irregular Graphs ◽  
Special Classes

: An k-edge-weighting of a graph G = (V, E) is a mapping : E(G) {1, 2, 3, ...k}, where k is a positive integer. The sum of the edge-weighting appearing on the edges incident at the vertex v under the edge-weighting and is denoted by (v). An k edge-weighting of G is a coprime irregular edge-weighting if gcd ( (u), (v)) = 1 for every pair of adjacent vertices u and v in G. A graph G admits a coprime irregular edge-weighting is called a coprime irregular graph. In this paper, we discuss the coprime irregular edge-weighting for some special classes of graphs.

scholarly journals More on Coprime Irregular Graphs

2019 ◽  
Vol 9 (1S4) ◽  
pp. 1014-1016
Keyword(s):  
Positive Integer ◽  
Edge Weighting ◽  
Irregular Graphs

An k−edge-weighting of a graph G = (V,E) is a map φ: E(G) → {1,2,3,...k}, where k is a positive integer. Denote Sφ(v) is the sum of edge-weights presenting on the edges incident at the vertex v under the edge-weighting φ. An k−edge-weighting of G is coprime irregular edge-weighting of G if gcd(Sφ(u),Sφ(v)) = 1 for every pair of adjacent vertices u and v in G. A graph G is coprime irregular if G admits a coprime irregular edge-weighting. In this paper, we discuss about coprime irregular edge-weighting for some families of graphs

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

Improving Color Constancy by Photometric Edge Weighting

2012 ◽  
Vol 34 (5) ◽  
pp. 918-929 ◽  
Author(s):  
A. Gijsenij ◽  
T. Gevers ◽  
J. van de Weijer
Keyword(s):  
Color Constancy ◽  
Edge Weighting

Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

2021 ◽  
pp. 1-20
Author(s):  
K. PUSHPA ◽  
K. R. VASUKI
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Divisor Function ◽  
Convolution Sums

Abstract The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals Repdigits as products of consecutive Padovan or Perrin numbers

2021 ◽  
Author(s):  
Salah Eddine Rihane ◽  
Alain Togbé
Keyword(s):  
Positive Integer ◽  
Decimal Expansion

AbstractA repdigit is a positive integer that has only one distinct digit in its decimal expansion, i.e., a number of the form $$a(10^m-1)/9$$ a ( 10 m - 1 ) / 9 , for some $$m\ge 1$$ m ≥ 1 and $$1 \le a \le 9$$ 1 ≤ a ≤ 9 . Let $$\left( P_n\right) _{n\ge 0}$$ P n n ≥ 0 and $$\left( E_n\right) _{n\ge 0}$$ E n n ≥ 0 be the sequence of Padovan and Perrin numbers, respectively. This paper deals with repdigits that can be written as the products of consecutive Padovan or/and Perrin numbers.

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