scholarly journals Graph Pebbling: A Blend of Graph Theory, Number Theory, and Optimization

2021 ◽  
Vol 68 (11) ◽  
pp. 1
Author(s):  
Glenn Hurlbert ◽  
Franklin Kenter
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Graph Pebbling

scholarly journals Nonexistence of Almost Moore Digraphs of Diameter Three

10.37236/811 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
J. Conde ◽  
J. Gimbert ◽  
J. Gonzàlez ◽  
J. M. Miret ◽  
R. Moreno
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Algebraic Number ◽  
Directed Graphs ◽  
Algebraic Number Theory ◽  
Cycle Structure ◽  
Spectral Techniques ◽  
Binary Matrices ◽  
Algebraic Multiplicities ◽  
Big Pieces

Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound $M(d,k)=1+d+\cdots +d^k$, where $d>1$ and $k>1$ denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when $d=2,3$ or $k=2$. In this paper, we prove that almost Moore digraphs of diameter $k=3$ do not exist for any degree $d$. The enumeration of almost Moore digraphs of degree $d$ and diameter $k=3$ turns out to be equivalent to the search of binary matrices $A$ fulfilling that $AJ=dJ$ and $I+A+A^2+A^3=J+P$, where $J$ denotes the all-one matrix and $P$ is a permutation matrix. We use spectral techniques in order to show that such equation has no $(0,1)$-matrix solutions. More precisely, we obtain the factorization in ${\Bbb Q}[x]$ of the characteristic polynomial of $A$, in terms of the cycle structure of $P$, we compute the trace of $A$ and we derive a contradiction on some algebraic multiplicities of the eigenvalues of $A$. In order to get the factorization of $\det(xI-A)$ we determine when the polynomials $F_n(x)=\Phi_n(1+x+x^2+x^3)$ are irreducible in ${\Bbb Q}[x]$, where $\Phi_n(x)$ denotes the $n$-th cyclotomic polynomial, since in such case they become 'big pieces' of $\det(xI-A)$. By using concepts and techniques from algebraic number theory, we prove that $F_n(x)$ is always irreducible in ${\Bbb Q}[x]$, unless $n=1,10$. So, by combining tools from matrix and number theory we have been able to solve a problem of graph theory.

Delving Deeper: The Chebyshev Polynomials: Patterns and Derivation

2004 ◽  
Vol 98 (1) ◽  
pp. 20-25 ◽  
Author(s):  
Benjamin Sinwell
Keyword(s):  
High School ◽  
Number Theory ◽  
Graph Theory ◽  
Fourier Series ◽  
High School Teachers ◽  
Chebyshev Polynomials ◽  
Mathematical Community ◽  
Russian Mathematician ◽  
School Teachers ◽  
Mathematical Exploration

Pafnuty Lvovich Chebyshev, a Russian mathematician, is famous for his work in the area of number theory and for his work on a sequence of polynomials that now bears his name. These Chebyshev polynomials have applications in the fields of polynomial approximation, numerical analysis, graph theory, Fourier series, and many other areas. They can be derived directly from the multiple-angle formulas for sine and cosine. They are relevant in high school and in the broader mathematical community. For this reason, the Chebyshev polynomials were chosen as one of the topics for study at the 2003 High School Teachers Program at the Park City Mathematics Institute (PCMI). The following is a derivation of the Chebyshev polynomials and a mathematical exploration of the patterns that they produce.

scholarly journals On the Possible Orders of a Basis for a Finite Cyclic Group

10.37236/351 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Peter Dukes ◽  
Peter Hegarty ◽  
Sarada Herke
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Cyclic Group ◽  
Additive Number Theory ◽  
Exact Results ◽  
Equivalent Problem ◽  
Additive Number ◽  
Finite Cyclic Group

We prove a result concerning the possible orders of a basis for the cyclic group ${\Bbb Z}_n$, namely: For each $k \in {\Bbb N}$ there exists a constant $c_k > 0$ such that, for all $n \in {\Bbb N}$, if $A \subseteq {\Bbb Z}_n$ is a basis of order greater than $n/k$, then the order of $A$ is within $c_k$ of $n/l$ for some integer $l \in [1,k]$. The proof makes use of various results in additive number theory concerning the growth of sumsets. Additionally, exact results are summarized for the possible basis orders greater than $n/4$ and less than $\sqrt{n}$. An equivalent problem in graph theory is discussed, with applications.

scholarly journals On a Connection of Number Theory with Graph Theory

2021 ◽  
pp. 121-139
Author(s):  
Michal Křížek ◽  
Lawrence Somer ◽  
Alena Šolcová
Keyword(s):  
Number Theory ◽  
Graph Theory

scholarly journals Search Heuristics and Constructive Algorithms for Maximally Idempotent Integers

Information ◽  
2021 ◽  
Vol 12 (8) ◽  
pp. 305
Author(s):  
Barry Fagin
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Positive Integer ◽  
Computer Science ◽  
Recent Work ◽  
Carmichael Numbers ◽  
Arbitrary Size ◽  
Explicit Formulation ◽  
Search Heuristics ◽  
Hard Problems

Previous work established the set of square-free integers n with at least one factorization n=p¯q¯ for which p¯ and q¯ are valid RSA keys, whether they are prime or composite. These integers are exactly those with the property λ(n)∣(p¯−1)(q¯−1), where λ is the Carmichael totient function. We refer to these integers as idempotent, because ∀a∈Zn,ak(p¯−1)(q¯−1)+1≡na for any positive integer k. This set was initially known to contain only the semiprimes, and later expanded to include some of the Carmichael numbers. Recent work by the author gave the explicit formulation for the set, showing that the set includes numbers that are neither semiprimes nor Carmichael numbers. Numbers in this last category had not been previously analyzed in the literature. While only the semiprimes have useful cryptographic properties, idempotent integers are deserving of study in their own right as they lie at the border of hard problems in number theory and computer science. Some idempotent integers, the maximally idempotent integers, have the property that all their factorizations are idempotent. We discuss their structure here, heuristics to assist in finding them, and algorithms from graph theory that can be used to construct examples of arbitrary size.

Simplicial Quadratic Forms

1983 ◽  
Vol 35 (1) ◽  
pp. 101-116 ◽  
Author(s):  
Barry Monson
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Hyperbolic Space ◽  
Quadratic Forms ◽  
Discrete Group ◽  
Unit Group ◽  
Fundamental Region ◽  
Preliminary Discussion ◽  
Klein Model ◽  
Extremal Properties

0. Introduction. Simplicial quadratic forms (cf. Definition 1.4), and various equivalent forms, have occasionally been studied in geometry [8], and in number theory [9], [10], in connection with the extremal properties of integral quadratic forms. Our investigations, which employ simple techniques from graph theory and geometry, partly continue both those of Coxeter [5], who introduced the graphs described in Section 1, and Vinberg [20], [21], who described an algorithm for determining a fundamental region for a discrete group acting on spherical, Euclidean, or hyperbolic space. After a preliminary discussion of reflexible forms and the Caley-Klein model for (n − 1)-space (1.2), we define a simplicial form and its graph. Having enumerated them completely, we turn in Section 2 to their equivalence, which is related to a geometric dissection. The unit group for each simplicial form can then be determined from Theorem 3.7.I wish to thank Professor H. S. M. Coxeter for many helpful ideas, and Professor G. Maxwell and the referee for suggesting numerous improvements.

scholarly journals Combinatorial Nullstellensatz

1999 ◽  
Vol 8 (1-2) ◽  
pp. 7-29 ◽  
Author(s):  
NOGA ALON
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Additive Number Theory ◽  
Graph Colouring ◽  
Combinatorial Nullstellensatz ◽  
Additive Number ◽  
Combinatorial Number Theory ◽  
Algebraic Technique

We present a general algebraic technique and discuss some of its numerous applications in combinatorial number theory, in graph theory and in combinatorics. These applications include results in additive number theory and in the study of graph colouring problems. Many of these are known results, to which we present unified proofs, and some results are new.

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