Multiple Divisor Research Articles

Characterisation of all integral circulant graphs with multiplicative divisor sets and few eigenvalues

AbstractWe present a method which in principal allows to characterise all integral circulant graphs with multiplicative divisor set having a spectrum, i.e. the set of distinct eigenvalues, of any given size. We shall exemplify the method for spectra of up to four eigenvalues, also reproving some known results for three eigenvalues along the way. In particular we show that given any integral circulant graph of arbitrary order n with multiplicative divisor set and precisely four distinct eigenvalues, n necessarily is either a prime power or the product of two prime powers with explicitly given simply structured divisor set and set of eigenvalues in both cases.

MMD labeling of complete tripartite graphs

A simple, connected and undirected graph with n vertices is said to have (MMD) modular multiplicative divisor labeling if there exist a one to one and on to function f from vertices of the graph to set of all natural numbers from 1 to n and label induced on the edges by the product of labels of end vertices modulo n such that addition of all edge labels is congruent to 0 (mod n). This paper studies MMD labeling of complete tripartite graphs and some open problems.

EASS of MMD graphs

Modular multiplicative divisor graph G with n vertices is a bijection g from V(G) to {1,2,…, n} and induced function h from edges of G to {0,1,…,n-1}, h(vw)=g(v)g(w) mod n for every vw in edges of G then adding the labels of all the edges is a multiple of n. Here studies MMD labeling of an even arbitrary supersubdivision (EASS) of cycle Cn .

بلاغة تناسب أسلوب القسم فی فواتح السور القرآنیة الحزب الأخیر نَموذجًا

الملخص تهدف الدراسة الإجابة عن أسئلة البحث، والتي أهمها: ما سر تعدد المقسم به تارة، وإفراده تارة أخرى؟ ولم آثر البيان القرآني لکل مقسم عليه مقسما به دون غيره؟ وما جهات التناسب بين المقسم به والمقسم عليه في فواتح سور الحزب الأخير؟ وما علاقة المقصد العام للسورة بجهات التناسب تلک؟ أما عن النتائج فأهمها: يحذف جواب القسم إذا اقتضاه المقام. أتى أسلوب القسم متماسکا، تمتد خلال سداه ولحمته خيوط التناسب، حتى يبدو محکم النسج، مما يجعل صورته المعنوية ـ والتي اکتساها أسلوب القسم ـ تقع في سمع المخاطب وروعه وقلبه کالکل المجتمع الذي تتداخل لبناته وتتماسک، وفي ذلک التأثير من الإمتاع والإقناع ما لا يخفى. في القسم جمع بين الأسلوبين الإنشائي بالقسم والخبري بجوابه، وفي ذلک ما لا يخفي من التأثير في النفس الإنسانية في المقامات والسياقات التي اقتضت ذلک. يقسم القرآن بالمحسوسات من الأمکنة والأزمنة والثمار والإنسان والحيوان على المعنويات والغيبيات، وفي ذلک من التوکيد والتأثير والإقناع بوقوع الغيبيات ما لا يخفى. أتى المقسم به المتعدد متناسبا بالتوافق أو التضاد حسب مقتضيات الأحوال. وقد اتبعت الدراسة المنهج: الوصفي التحليلي. Summary The study aims to answer the research questions, the most important of which are: What is the secret of the multiplicity of the denominator at one time, and the individual at other times? Did the effect of the Qur’anic statement for each one divided on it divided by another? What are the points of proportionality between the divisor and the divisor in the opening of the last party fence? What is the relationship of the general purpose of the surah to those bodies of proportionality? As for the results, the most important of them are: The oath of the department is deleted if required by the station. The style of the department was coherent, extending through its streak and its flesh the threads of proportionality, until it appears tightly woven, which makes its moral image - and that it covered the style of the section - lies in the hearing of the addressee, his terror, and his heart like the whole society whose daughters overlap and hold together, and in that effect of enjoyment and persuasion is not hidden. In the section, he combined the structural methods of the department and the expert with his answer, and in that he does not conceal from the influence on the human soul in the positions and contexts that required that. The Qur’an is divided with perceptions from places, times, fruits, people and animals, according to morale and occultation, and in that affirmation, influence and persuasion of the occurrence of the occult is what is not hidden. The multiple divisor is proportional to compatibility or contradiction, as appropriate. The study followed the method: descriptive analytical.

The Effect of Guided Inquiry Learning Model to the Metacognitive Ability of Primary School Students

This study aimed at analyzing the effect of guided inquiry learning to the metacognitive ability of primary school students on the material of Least Common Multiple (KPK) and Greatest Common Divisor (FPB). The type of study was a mixed-method using quantitative and qualitative methods. There were 55 students of 4th grade used as the subjects of study. Two learning models were compared, namely guided inquiry learning model and conventional learning model. The students’ metacognitive ability was measured by means of problem-solving test on the material of Least Common Multiple (KPK) and Greatest Common Divisor (FPB). The quantitative analysis data used descriptive and inferential statistical tests. According to the results of data analysis, it was discovered that the t-test of sig (2-tailed) from the independent samples t-test of post-test was 0,00 (p = <0,05); this indicated that there was a significant difference on it. This showed that there was a difference of students’ metacognitive ability for both classes in solving the problems of Least Common Multiple (KPK) and Greatest Common Divisor (FPB) after the guided inquiry learning was implemented. Consequently, it can be concluded that there is a significant effect on the implementation of guided inquiry learning model to improve the students’ metacognitive ability in solving the material problems of Least Common Multiple (KPK) and Greatest Common Divisor (FPB).

EASS OF CYCLE RELATED MMD GRAPHS

Adv. Math. Sci. J. Vol. 9, No. 3 Open Access EASS OF CYCLE RELATED MMD GRAPHS R. Revathi and R. Mary Jeya Jothi DOI: https://doi.org/10.37418/amsj.9.3.75 Full Text Abstract Abstract A simple, connected and undirected graph with vertices is said to have (MMD) modular multiplicative divisor labeling if there exists a one to … page AMSJ9-N3-75 Read More »

Symmetries in A-type little string theories. Part II. Eisenstein series and generating functions of multiple divisor sums

We continue our study of symmetries of a class of little string theories of A-type, which are engineered by N parallel M5-branes probing a flat transverse space. Extending the analysis of the companion paper [1], we discuss the part of the free energy that is sensitive to the details of the \U0001d51eN−1 gauge structure, by computing explicit series expansions for the cases N = 2, 3, 4. Based on these examples, we find a class of functions that we conjecture to resum whole sectors in the instanton expansion of the free energy and which combine in a natural manner its modular properties as well as the gauge symmetry. These functions have previously been introduced in the literature as the generating functions of multi-divisor sums and in the case N = 2 can also be cast into the form of a generalised Eisenstein series. We use these resummed contributions to the free energy to perform a number of non-trivial consistency checks for our results.

On Fujita’s conjecture for pseudo-effective thresholds

We show Fujita's spectrum conjecture for $\epsilon$-log canonical pairs and Fujita's log spectrum conjecture for log canonical pairs. Then, we generalize the pseudo-effective threshold of a single divisor to multiple divisors and establish the analogous finiteness and the DCC properties.

Dakota (Dakon Matematika) Sebagai Media Penanaman Konsep KPK dan FBB di Sekolah Dasar

Mathematics teaching nowadays still requires students to memorize formulas but does not provide an understanding of concepts.Teachers who only teaches mathematics monotonously will result in students’ learning motivation decreasing. They need to explore the use of media to help students experiencing a meaningful and practical concept. This study tried to explore the potential of Dakon traditional game as media to encourage students’ motivation and concept understanding. Dakon is proposed as a media to help students understand the material conceptually by fun learning. Dakon Matematika (Dakota) is presented as one of the media that allows students in understanding Least Common Multiple (KPK) and Greatest Common Divisor (FPB) concept in elementary school. Dakon Matematika (Dakota) potentially can foster student learning motivation because besides studying mathematics it also introduces students to one of the Indonesian traditional games that are almost abandoned. This result of study promoted the Dakon and other traditional games to in mathematic learning. Some research on the use of Dakon in mathematics learning also showed the potential use of Dakon Matematikagames in learning mathematics, especially in the KPK and FPB material at elementary school level.

Structural behaviour of MMD labeling on some SSP bipartite graphs

A graph G(V, E) with n vertices is said to have modular multiplicative divisor (MMD) labeling if there exist a bijection f:V(G) → {1, 2,…,n} and the induced function f*:E(G) → {0, 1, 2,…,n − 1}where f*(uv) = f(u)f(v)(mod n) for all uv ∊ E(G) such that n divides the sum of all edge labels of G. The graph G is super strongly perfect (SSP) if every induced subgraph H of G contains a minimal dominating set that meets all the maximal cliques of H. This paper investigates some families of SSP bipartite graph structure using MMD labeling.

On the kernel of integral circulant graphs

Given a positive integer n, every integral circulant graph on n vertices is isomorphic to some graph ICG(n,D) having vertex set Z/nZ and edge set {(a,b):a,b∈Z/nZ,gcd⁡(a−b,n)∈D} for a uniquely determined set D of positive divisors of n. By virtue of its adjacency matrix, one defines the spectrum of a graph G and, naturally, can ask to which degree the eigenvalues of G determine the graph itself. With respect to integral circulant graphs little is known about this question, which is related to a conjecture of So.In this note we examine the role of the eigenvalue 0 and clarify the interrelation between the dimension of the kernel of ICG(n,D) and the graph itself for all prime powers n=pk and for all positive integers n in case D is a multiplicative divisor set.

MULTIPLE CO-PRIME DIVISORS AND MULTIPLE DIVISORS GRAPHS

In this paper we introduce two new concepts of graphs. Let R be a commutative ring with identity and Z(R)\{0} be the set of elements in R divided it into two sets Z1(R) and Z2(R) , where Z1(R) be the set of co-prime divisor elements satisfies the Euler’s function and Z2(R) be the set of non-co-prime divisor elements, a simple graph MC(R) is associated to Z(R) it is called multiple co-prime divisors graph. Moreover, Let R be a commutative ring with identity and Z*(R)=Z(R)\{0} be the set of all non-zero divisor elements in R, a simple graph M(R) is associated to R for distinct elements a and b in Z*(R) is an edge in M(R) if and only if aba=a(mod n) for all n in N. Also, diameter, girth, chromatic number and nullity of multiple co-prime divisor and multiple divisor graphs will be determined.

Modular Multiplicative Divisor Labeling of k-Multilevel Corona Related Graphs

A graph G(V, E) with n vertices is said to have modular multiplicative divisor labeling if there exist a bijection f: V(G) → {1,2,… n} and the induced function f*: E(G) → {0,1,2,…, n − 1} where f*(uv) = f(u)f(V) mod n for all uv ∈ E(G) such that n divides the sum of all edge labels of G. In this paper we discuss new results on modular multiplicative divisor (abbreviated as MMD) labeling of k-multilevel corona of path P3 with disjoint union of 2m copies of k1 and k-multilevel corona of Cn+ (adding pendant edge for each vertex of the cycle Cn) with disjoint union of 2m copies of path P3, where k and m are positive integers.

A Curious Result for GCDs and LCMs

SummaryWe give necessary and sufficient conditions under which the sum of two positive integers equals the difference of their least common multiple and greatest common divisor, and then prove this result.

Characterization of certain families of modular multiplicative divisor graphs

A modular multiplicative divisor (MMD) labelling of a graph G with p vertices and q edges is a bijection f from the set of all vertices to the set of positive integers {1, 2,…,p} such that the sum of all edge labels f*(uv)=f(u)f(v)(modp); uv∈E(G) is a multiple of p. In this paper, we characterize certain classes of MMD graphs. This result is a solution to the open problem posed by G. Sethuraman in AKCE International Journal of Graphs and Combinatorics, No. 1 (2009) 229–236. In addition, we discuss the role of MMD labelling in obtaining cyclic decompositions of certain families of graphs into specific graphs.

Double shuffle relations of double zeta values and the double Eisenstein series at levelN

In their seminal paper "Double zeta values and modular forms" Gangl, Kaneko and Zagier defined a double Eisenstein series and used it to study the relations between double zeta values. One of their key ideas is to study the formal double space and apply the double shuffle relations. They also proved the double shuffle relations for the double Eisenstein series. More recently, Kaneko and Tasaka extended the double Eisenstein series to level 2, proved its double shuffle relations and studied the double zeta values of level 2. Motivated by the above works, we define in this paper the corresponding objects at higher levels and prove that the double Eisenstein series of level N satisfies the double shuffle relations for every positive integer N. In order to obtain our main theorem we prove a key result on the multiple divisor functions of level N and then use it to solve a complicated under-determined system of linear equations by some standard techniques from linear algebra.

The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

We study the algebra $${{\mathrm{{\mathcal {MD}}}}}$$ of generating functions for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in $${\mathbb {Q}}$$ arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra $${{\mathrm{{\mathcal {MD}}}}}$$ is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in $${{\mathrm{{\mathcal {MD}}}}}$$ . The (quasi-)modular forms for the full modular group $${{\mathrm{SL}}}_2({\mathbb {Z}})$$ constitute a subalgebra of $${{\mathrm{{\mathcal {MD}}}}}$$ , and this also yields linear relations in $${{\mathrm{{\mathcal {MD}}}}}$$ . Generating functions of multiple divisor sums can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those of length 2, coming from modular forms.

Integral circulant Ramanujan graphs via multiplicativity and ultrafriable integers

Any integral circulant graph ICG(n,D) is characterised by its order n and a set D of positive divisors of n in such a way that it has vertex set Z/nZ and edge set {(a,b):a,b∈Z/nZ,gcd⁡(a−b,n)∈D}. Such graphs are regular, and a connected ρ-regular graph G is called Ramanujan if the second largest modulus of the eigenvalues of the adjacency matrix of G is at most 2ρ−1.In 2010 Droll described all Ramanujan unitary Cayley graphs, i.e. graphs of type Xn:=ICG(n,{1}) having the Ramanujan property. Recently, Le and the author classified all Ramanujan graphs ICG(ps,D) for prime powers ps and arbitrary divisor sets D. We greatly extend the established results to graphs ICG(n,D) with arbitrary n and multiplicative divisor set D: (i) We derive a criterion (in terms of Euler's totient function) for ICG(n,D) to be Ramanujan. (ii) We prove that for all even integers n>2 and a positive proportion of the odd integers n, namely those having a “dominating” prime power factor, there exists a multiplicative divisor set D such that ICG(n,D) is Ramanujan. (iii) We show that the set of odd n for which no Ramanujan graph ICG(n,D) with multiplicative divisor set D exists, viz. ultrafriable integers, has positive density as well. The proofs of (ii) and (iii) use methods from analytic number theory.

On So's conjecture for integral circulant graphs

Each integral circulant graph ICG(n,D) is characterised by its order n and a set D of positive divisors of n in such a way that it has vertex set Z=nZ and edge set {(a,b) : a, b ? Z=nZ, gcd(a - b,n) ? D}. According to a conjecture of So two integral circulant graphs are isomorphic if and only if they are isospectral, i.e. they have the same eigenvalues (counted with multiplicities). We prove a weaker form of this conjecture, namely, that two integral circulant graphs with multiplicative divisor sets are isomorphic if and only if their spectral vectors coincide.

Modular Multiplicative Divisor Labeling of Some Path Related Graphs

A graph G (V,E) with |V| = n is said to have modular multiplicative divisor labeling if there exist a bijection f: V(G) → {1, 2, …,} and the induced function f * : E(G) → {0, 1, 2, …, n - 1} where f(uv) = f(u)f(v) (mod n) such that n divides the sum of all edge labels of G. We prove that the path P n , and the graph P a, b (a graph which connects two vertices by means of "b" internally disjoint paths of length "a" each), shadow graph of a path and the cartesian product P n × P 1 , (n is not a multiple of 6) admits modular multiplicative divisor labeling. Also we discuss the upper bound for the number of edges in a modular multiplicative divisor graphs. AMS Subject Classification: 05C78