scholarly journals Two coloring problems on matrix graphs

2016 ◽  
Vol 08 (03) ◽  
pp. 1650053
Author(s):  
Zhe Han ◽  
Mei Lu
Keyword(s):  
Finite Field ◽  
Optical Networks ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Basic Properties ◽  
New Family ◽  
Minimum Number ◽  
Vertex Set ◽  
Positive Integers

In this paper, we propose a new family of graphs, matrix graphs, whose vertex set [Formula: see text] is the set of all [Formula: see text] matrices over a finite field [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text]. And any two matrices share an edge if the rank of their difference is [Formula: see text]. Next, we give some basic properties of such graphs and also consider two coloring problems on them. Let [Formula: see text] (resp., [Formula: see text]) denote the minimum number of colors necessary to color the above matrix graph so that no two vertices that are at a distance at most [Formula: see text] (resp., exactly [Formula: see text]) get the same color. These two problems were proposed in the study of scalability of optical networks. In this paper, we determine the exact value of [Formula: see text] and give some upper and lower bounds on [Formula: see text].

scholarly journals Wiener–Hosoya Matrix of Connected Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 359
Author(s):  
Hassan Ibrahim ◽  
Reza Sharafdini ◽  
Tamás Réti ◽  
Abolape Akwu
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Wiener Index ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Connected Graphs ◽  
Matrix Description ◽  
Vertex Set

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

scholarly journals Expected Logarithm and Negative Integer Moments of a Noncentral χ2-Distributed Random Variable

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 1048
Author(s):  
Stefan Moser
Keyword(s):  
Closed Form ◽  
Lower Bounds ◽  
Degrees Of Freedom ◽  
Random Variable ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Basic Properties

Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral χ2-distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed.

2-Distance chromatic number of some graph products

2020 ◽  
Vol 12 (02) ◽  
pp. 2050021
Author(s):  
Ghazale Ghazi ◽  
Freydoon Rahbarnia ◽  
Mostafa Tavakoli
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Graph Products ◽  
Lexicographic Product ◽  
Graph Product ◽  
Minimum Number

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

Almost Safe Group Testing with Few Tests

1994 ◽  
Vol 3 (3) ◽  
pp. 411-419
Author(s):  
Andrzej Pelc
Keyword(s):  
Lower Bounds ◽  
Group Testing ◽  
Upper And Lower Bounds ◽  
Probability Functions ◽  
Minimum Number ◽  
Location Strategies

In group testing, sets of data undergo tests that reveal if a set contains faulty data. Assuming that data items are faulty with given probability and independently of one another, we investigate small families of tests that enable us to locate correctly all faulty data with probability converging to one as the amount of data grows. Upper and lower bounds on the minimum number of such tests are established for different probability functions, and respective location strategies are constructed.

scholarly journals On numbers n dividing the nth term of a linear recurrence

2012 ◽  
Vol 55 (2) ◽  
pp. 271-289 ◽  
Author(s):  
Juan José Alba González ◽  
Florian Luca ◽  
Carl Pomerance ◽  
Igor E. Shparlinski
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Linear Recurrence ◽  
Recurrent Sequence ◽  
Positive Integers

AbstractWe give upper and lower bounds on the count of positive integers n ≤ x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.

Upper and Lower Bounds for a Function Related to Brown's Lemma

Integers ◽  
2010 ◽  
Vol 10 (6) ◽  
Author(s):  
Hayri Ardal
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Fixed Positive Integer ◽  
Positive Integers

AbstractThe well-known Brown's lemma says that for every finite coloring of the positive integers, there exist a fixed positive integer

scholarly journals REPARTITORS, SELECTORS AND SUPERSELECTORS

2006 ◽  
Vol 07 (03) ◽  
pp. 391-415 ◽  
Author(s):  
FRÉDÉRIC HAVET
Keyword(s):  
Lower Bounds ◽  
Disjoint Paths ◽  
Edge Disjoint ◽  
Minimum Number ◽  
Vertex Set ◽  
Optimal Networks

An (n, p, f)-network G is a graph (V, E) where the vertex set V is partitioned into four subsets [Formula: see text] and [Formula: see text] called respectively the priorities, the ordinary inputs, the outputs and the switches, satisfying the following constraints: there are p priorities, n - p ordinary inputs and n + f outputs; each priority, each ordinary input and each output is connected to exactly one switch; switches have degree at most 4. An (n, p, f)-network is an (n, p, f)-repartitor if for any disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] with [Formula: see text] and [Formula: see text], there exist in G, n edge-disjoint paths, p of them from [Formula: see text] to [Formula: see text] and the n - p others joining [Formula: see text] to [Formula: see text]. The problem is to determine the minimum number R(n, p, f) of switches of an (n, p, f)-repartitor and to construct a repartitor with the smallest number of switches. In this paper, we show how to build general repartitors from (n, 0, f)-repartitors also called (n, n + f)-selectors. We then consrtuct selectors using more powerful networks called superselectors. An (n, 0, 0)-network is an n-superselector if for any subsets [Formula: see text] and [Formula: see text] with [Formula: see text], there exist in G, [Formula: see text] edge-disjoint paths joining [Formula: see text] to [Formula: see text]. We show that the minimum number of switches of an n-superselector S+ (n) is at most 17n + O(log(n)). We then deduce that [Formula: see text] if [Formula: see text], R(n, p, f) ≤ 18n + 34f + O( log (n + f)), if [Formula: see text] and [Formula: see text] if [Formula: see text]. Finally, we give lower bounds for R(n, 0, f) and S+ (n) and show optimal networks for small value of n.

The Thickness of the Cartesian Product of Two Graphs

2016 ◽  
Vol 59 (4) ◽  
pp. 705-720
Author(s):  
Yichao Chen ◽  
Xuluo Yin
Keyword(s):  
Lower Bounds ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
Minimal Graph ◽  
Minimum Number ◽  
Complete Bipartite Graphs ◽  
Planar Subgraphs ◽  
Proper Subgraph

AbstractThe thickness of a graph G is the minimum number of planar subgraphs whose union is G. A t-minimal graph is a graph of thickness t that contains no proper subgraph of thickness t. In this paper, upper and lower bounds are obtained for the thickness, t(G ⎕ H), of the Cartesian product of two graphs G and H, in terms of the thickness t(G) and t(H). Furthermore, the thickness of the Cartesian product of two planar graphs and of a t-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph K4k,4k, the thickness of the Cartesian product of two complete bipartite graphs Kn,n and Kn,n is also given for n≠4k + 1.

COVERING A SET OF POINTS WITH A MINIMUM NUMBER OF TURNS

2004 ◽  
Vol 14 (01n02) ◽  
pp. 105-114 ◽  
Author(s):  
MICHAEL J. COLLINS
Keyword(s):  
Euclidean Space ◽  
Lower Bounds ◽  
Piecewise Linear ◽  
Upper And Lower Bounds ◽  
Linear Path ◽  
Change Direction ◽  
Finite Set ◽  
Minimum Number ◽  
Pass Through ◽  
Set Of Points

Given a finite set of points in Euclidean space, we can ask what is the minimum number of times a piecewise-linear path must change direction in order to pass through all of them. We prove some new upper and lower bounds for the rectilinear version of this problem in which all motion is orthogonal to the coordinate axes. We also consider the more general case of arbitrary directions.

scholarly journals COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS

2013 ◽  
Vol 94 (1) ◽  
pp. 50-105 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ ◽  
TERENCE TAO
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Diophantine Equation ◽  
Upper And Lower Bounds ◽  
Natural Numbers ◽  
Number Of Solutions ◽  
Waring’S Problem ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

AbstractFor any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$ with $x, y, z$ positive integers. The Erdős–Straus conjecture asserts that $f(n)\gt 0$ for every $n\geq 2$. In this paper we obtain a number of upper and lower bounds for $f(n)$ or $f(p)$ for typical values of natural numbers $n$ and primes $p$. For instance, we establish that $$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$ These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.

Sign in / Sign up

Export Citation Format

Share Document