scholarly journals NOTE ON SUPER \((a,1)\)–\(P_3\)–ANTIMAGIC TOTAL LABELING OF STAR \(S_n\)

2021 ◽  
Vol 7 (2) ◽  
pp. 86
Author(s):  
S. Rajkumar ◽  
M. Nalliah ◽  
Madhu Venkataraman
Keyword(s):  
Arithmetic Progression ◽  
Simple Graph ◽  
Positive Integers

Let \(G=(V, E)\) be a simple graph and \(H\) be a subgraph of \(G\). Then \(G\) admits an \(H\)-covering, if every edge in \(E(G)\) belongs to at least one subgraph of \(G\) that is isomorphic to \(H\). An \((a,d)-H\)-antimagic total labeling of \(G\) is bijection \(f:V(G)\cup E(G)\rightarrow \{1, 2, 3,\dots, |V(G)| + |E(G)|\}\) such that for all subgraphs \(H'\) of \(G\) isomorphic to \(H\), the \(H'\) weights \(w(H') =\sum_{v\in V(H')} f (v) + \sum_{e\in E(H')} f (e)\) constitute an arithmetic progression \(\{a, a + d, a + 2d, \dots , a + (n- 1)d\}\), where \(a\) and \(d\) are positive integers and \(n\) is the number of subgraphs of \(G\) isomorphic to \(H\). The labeling \(f\) is called a super \((a, d)-H\)-antimagic total labeling if \(f(V(G))=\{1, 2, 3,\dots, |V(G)|\}.\) In [5], David Laurence and Kathiresan posed a problem that characterizes the super \( (a, 1)-P_{3}\)-antimagic total labeling of Star \(S_{n},\) where \(n=6,7,8,9.\)  In this paper, we completely solved this problem.

scholarly journals The Irregularity and Modular Irregularity Strength of Fan Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 605
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Marcela Lascsáková ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Isolated Vertex ◽  
Positive Integers ◽  
Irregularity Strength

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

scholarly journals SUMS OF PRODUCTS OF CONGRUENCE CLASSES AND OF ARITHMETIC PROGRESSIONS

2009 ◽  
Vol 05 (04) ◽  
pp. 625-634
Author(s):  
SERGEI V. KONYAGIN ◽  
MELVYN B. NATHANSON
Keyword(s):  
Arithmetic Progression ◽  
Congruence Class ◽  
Arithmetic Progressions ◽  
Sum Of Products ◽  
Sums Of Products ◽  
Positive Integers ◽  
Congruence Classes

Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a + im : i ∈ N0}. For positive integers a,b,c,d,m the sum of products set Rm(a)Rm(b) + Rm(c)Rm(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd (a,b,c,d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab+cd), and that the sum of products set Pm(a)Pm(b)+Pm(c)Pm eventually coincides with the infinite arithmetic progression Pm(ab+cd).

scholarly journals Sums of powers of integers and hyperharmonic numbers

2021 ◽  
Vol 27 (2) ◽  
pp. 101-110
Author(s):  
José Luis Cereceda
Keyword(s):  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Explicit Representation ◽  
Sums Of Powers ◽  
Hyperharmonic Numbers ◽  
New Formula ◽  
Positive Integers

In this paper, we obtain a new formula for the sums of k-th powers of the first n positive integers, Sk(n), that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Furthermore, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for Sk(n) to negative values of n.

scholarly journals Another Antimagic Decomposition of Generalized Peterzen Graph

2021 ◽  
Vol 3 (2) ◽  
pp. 92-100
Author(s):  
Nur Inayah
Keyword(s):  
Arithmetic Progression ◽  
Positive Integers

AbstractA decomposition of a graph P into a family Q consisting of isomorphic copies of a graph Q is (a,b)-Q-antimagic if there is a bijection φ:V(P)∪E(P)→{1,2,3,4…,v_P+e_P} such that for all subgraphs Q’ isomorphic to Q,   the Q-weightsφ(Q’ )=∑_(v∈V(Q^' ))▒φ(v) + ∑_(e∈E(Q^'))▒〖φ(e)〗constitute an arithmetic progression a,a + b,a + 2b,…,a + (r - 1)b where a and b are positive integers and r is the number of subgraphs of P isomorphic to Q. In this article, we prove the existence of a (a,b)-P_4-antimagic  decomposition of a generalized Peterzen graph GPz(n,3) for several values of b.Keywords: covering; decomposition; antimagic; generalized Peterzen. AbstrakSuatu dekomposisi dari suatu graf P ke dalam suatu famili Q yang terdiri dari salinan isomorfik dari graf Q dikatakan (a,b)-Q-antiajaib jika terdapat pemetaaan bijektif φ:V(P)∪E(P)→{1,2,3,4…,v_P+e_P} sedemikian sehingga semua subgraf Q’ yang isomorfik ke Q, dengan bobot-Q sebagai berikutφ(Q’ )=∑_(v∈V(Q^' ))▒φ(v) + ∑_(e∈E(Q^'))▒〖φ(e)〗yang membentuk suatu barisan aritmatika yaitu a,a + b,a + 2b,…,a + (r - 1)b dengan a dan b adalah bilangan bulat positif dan r adalah banyaknya subgraf dari P yang isomorfik ke Q. Pada artikel ini, kami membuktikan eksistensi (a,b)-P_4-antiajaib dekomposisi dari graf generalized Peterzen GPz(n,3) untuk beberapa nilai b.Kata kunci: selimut; dekomposisi; antiajaib; generalized Peterzen.

scholarly journals On the Total Vertex Irregular Labeling of Proper Interval Graphs

2020 ◽  
Vol 12 (4) ◽  
pp. 537-543
Author(s):  
A. Rana
Keyword(s):  
Efficient Algorithm ◽  
Simple Graph ◽  
Interval Graphs ◽  
Vertex Weight ◽  
Vertex Set ◽  
Positive Integers ◽  
Irregularity Strength

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers).  For a simple graph G = (V, E) with vertex set V and edge set E, a labeling  Φ: V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x∈ V under a total k-labeling  Φ is defined as wt(x) = Φ(x) + ∑y∈N(x) Φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is defined to be a vertex irregular total labeling of a graph, if for every two different vertices x and y of G, wt(x)≠wt(y). The minimum k for which  a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs is presented for interval graphs.

On Sums of Progressions of Positive Integers

1970 ◽  
Vol 54 (388) ◽  
pp. 113-115
Author(s):  
R. L. Goodstein
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Common ◽  
Necessary And Sufficient ◽  
Positive Integers

We consider the problem of finding necessary and sufficient conditions for a positive integer to be the sum of an arithmetic progression of positive integers with a given common difference, starting with the case when the common difference is unity.

The Average Number of Divisors in an Arithmetic Progression

1981 ◽  
Vol 24 (1) ◽  
pp. 37-41 ◽  
Author(s):  
R. A. Smith ◽  
M. V. Subbarao
Keyword(s):  
Arithmetic Progression ◽  
Divisor Function ◽  
Number Of Divisors ◽  
Image Position ◽  
Positive Integers

Let l and k be positive integers. Then for each integer n ≥ 1, define d(n; l, k) to be the number of (positive) divisors of n which lie in the arithmetic progression I mod k. Note that d(n;1,1) = d(n), the ordinary divisor function.

Notes on an Extension of Pythagorean Triplets in Arithmetic Progression

1969 ◽  
Vol 62 (8) ◽  
pp. 633-635
Author(s):  
William M. Waters
Keyword(s):  
Arithmetic Progression ◽  
All Solutions ◽  
The Common ◽  
Positive Integers

IF ONE considers the equation x2 + y2 = z2, where x, y, and z are positive integers in arithmetic progression, it is obvious that the primitive solution of this equation is {3, 4, 5} and that all solutions are of the form {3d, 4d, 5d}, where d represents the common difference of the arithmetic progression.

scholarly journals Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

2014 ◽  
Vol 57 (3) ◽  
pp. 551-561 ◽  
Author(s):  
Daniel M. Kane ◽  
Scott Duke Kominers
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Finite Arithmetic ◽  
Positive Integers

AbstractFor relatively prime positive integers u0 and r, we consider the least common multiple Ln := lcm(u0, u1..., un) of the finite arithmetic progression . We derive new lower bounds on Ln that improve upon those obtained previously when either u0 or n is large. When r is prime, our best bound is sharp up to a factor of n + 1 for u0 properly chosen, and is also nearly sharp as n → ∞.

On (k, l)-sets in cyclic groups of odd prime order

2001 ◽  
Vol 63 (1) ◽  
pp. 115-121 ◽  
Author(s):  
T. Bier ◽  
A. Y. M. Chin
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Arithmetic Progression ◽  
Prime Order ◽  
Finite Abelian Group ◽  
Maximum Cardinality ◽  
Cyclic Groups ◽  
Positive Integers

Let A be a finite Abelian group written additively. For two positive integers k, l with k ≠ l, we say that a subset S ⊂ A is of type (k, l) or is a (k, l) -set if the equation x1 + x2 + … + xk − xk+1−… − xk+1 = 0 has no solution in the set S. In this paper we determine the largest possible cardinality of a (k, l)-set of the cyclic group ℤP where p is an odd prime. We also determine the number of (k, l)-sets of ℤp which are in arithmetic progression and have maximum cardinality.

Sign in / Sign up

Export Citation Format

Share Document