scholarly journals Optimal L(h,k)-Labeling of Regular Grids

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Tiziana Calamoneri
Keyword(s):  
Lower Bounds ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Vertex Coloring ◽  
Small Interval ◽  
Regular Grids ◽  
Special Cases ◽  
Hexagonal Grids ◽  
International Audience ◽  
The Difference

International audience The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that 'close' nodes have labels which differ by at least k, and 'very close' nodes have labels which differ by at least h. The span of an L(h,k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeking those with minimum span for each value of k and h ≥ k. The L(h,k)-labeling problem has been intensively studied in some special cases, i.e. when k=0 (vertex coloring), h=k (vertex coloring the square of the graph) and h=2k (radio- or λ -coloring) but no results are known in the general case for regular grids. In this paper, we completely solve the L(h,k)-labeling problem on regular grids, finding exact values of the span for each value of h and k; only in a small interval we provide different upper and lower bounds.

scholarly journals Minimum survivable graphs with bounded distance increase

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Selma Djelloul ◽  
Mekkia Kouider
Keyword(s):  
Fault Tolerance ◽  
Lower Bounds ◽  
Interconnection Networks ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Minimum Size ◽  
Np Hard ◽  
Bounded Distance ◽  
Minimum Number ◽  
International Audience

International audience We study in graphs properties related to fault-tolerance in case a node fails. A graph G is k-self-repairing, where k is a non-negative integer, if after the removal of any vertex no distance in the surviving graph increases by more than k. In the design of interconnection networks such graphs guarantee good fault-tolerance properties. We give upper and lower bounds on the minimum number of edges of a k-self-repairing graph for prescribed k and n, where n is the order of the graph. We prove that the problem of finding, in a k-self-repairing graph, a spanning k-self-repairing subgraph of minimum size is NP-Hard.

Optimal L(j,k)-Edge-Labeling of Regular Grids

2015 ◽  
Vol 26 (04) ◽  
pp. 523-535 ◽  
Author(s):  
Tiziana Calamoneri
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Regular Grids ◽  
Triangular Grids ◽  
Minimum Value ◽  
Special Cases ◽  
Common Endpoint ◽  
Positive Integers ◽  
Edge Labeling

Given a graph [Formula: see text] and two positive integers j and k, an [Formula: see text]-edge-labeling is a function f assigning to edges of E colors from a set [Formula: see text] such that [Formula: see text] if e and e′ are adjacent, i.e. they share a common endpoint, [Formula: see text] if e and e′ are not adjacent and there exists an edge adjacent to both e and e′. The aim of the [Formula: see text]-edge-labeling problem consists of finding a coloring function f such that the value of [Formula: see text] is minimum. This minimum value is called [Formula: see text]. This problem has already been studied on hexagonal, squared and triangular grids, but mostly not coinciding upper and lower bounds on [Formula: see text] have been proposed. In this paper we close some of these gaps or find better bounds on [Formula: see text] in the special cases [Formula: see text] and [Formula: see text]. Moreover, we propose tight [Formula: see text]-edge-labelings for eight-regular grids.

scholarly journals Refinements of the Lower Bounds of the Jensen Functional

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Iva Franjić ◽  
Sadia Khalid ◽  
Josip Pečarić
Keyword(s):  
Lower Bounds ◽  
Jensen Inequality ◽  
Left Hand ◽  
Right Hand ◽  
Special Cases ◽  
Jensen Functional ◽  
The Difference ◽  
The Right

The lower bounds of the functional defined as the difference of the right-hand and the left-hand side of the Jensen inequality are studied. Refinements of some previously known results are given by applying results from the theory of majorization. Furthermore, some interesting special cases are considered.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals Positive solutions and eigenvalues of conjugate boundary value problems

1999 ◽  
Vol 42 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Lower Bounds ◽  
Boundary Value ◽  
Upper And Lower Bounds ◽  
Special Cases ◽  
Image Position

We consider the following boundary value problemwhere λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

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.

scholarly journals On the connectedness and diameter of a geometric Johnson graph

2013 ◽  
Vol Vol. 15 no. 3 (Combinatorics) ◽  
Author(s):  
Crevel Bautista-Santiago ◽  
Javier Cano ◽  
Ruy Fabila-Monroy ◽  
David Flores-Peñaloza ◽  
Hernàn González-Aguilar ◽  
...  
Keyword(s):  
Lower Bounds ◽  
General Position ◽  
Convex Set ◽  
Upper And Lower Bounds ◽  
Johnson Graph ◽  
International Audience

Combinatorics International audience Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P \C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough values of n, this graph is connected, and give upper and lower bounds on its diameter.

scholarly journals Spectral Properties with the Difference between Topological Indices in Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Akbar Jahanbani ◽  
Roslan Hasni ◽  
Zhibin Du ◽  
Seyed Mahmoud Sheikholeslami
Keyword(s):  
Characteristic Polynomial ◽  
Lower Bounds ◽  
Adjacency Matrix ◽  
Spectral Properties ◽  
Main Tool ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
New Energy ◽  
Energy Of Graph ◽  
The Difference

Let G be a graph of order n with vertices labeled as v1,v2,…,vn. Let di be the degree of the vertex vi, for i=1,2,…,n. The difference adjacency matrix of G is the square matrix of order n whose i,j entry is equal to di+dj−2−1/didj if the vertices vi and vj of G are adjacent or vivj∈EG and zero otherwise. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix, that is, a modification of the classical adjacency matrix involving the degrees of the vertices. In this paper, some properties of its characteristic polynomial are studied. We also investigate the difference energy of a graph. In addition, we establish some upper and lower bounds for this new energy of graph.

ON THE DIFFERENCE OF COEFFICIENTS OF OZAKI CLOSE-TO-CONVEX FUNCTIONS

2020 ◽  
pp. 1-8
Author(s):  
YOUNG JAE SIM ◽  
DEREK K. THOMAS
Keyword(s):  
Unit Disk ◽  
Lower Bounds ◽  
Convex Functions ◽  
Univalent Functions ◽  
Upper And Lower Bounds ◽  
The Difference

Let $f$ be analytic in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ and ${\mathcal{S}}$ be the subclass of normalised univalent functions given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ for $z\in \mathbb{D}$ . We give sharp upper and lower bounds for $|a_{3}|-|a_{2}|$ and other related functionals for the subclass ${\mathcal{F}}_{O}(\unicode[STIX]{x1D706})$ of Ozaki close-to-convex functions.

scholarly journals EXACT UPPER AND LOWER BOUNDS ON THE DIFFERENCE BETWEEN THE ARITHMETIC AND GEOMETRIC MEANS

2015 ◽  
Vol 92 (1) ◽  
pp. 149-158 ◽  
Author(s):  
IOSIF PINELIS
Keyword(s):  
Convex Function ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Jensen Inequality ◽  
Geometric Means ◽  
The Difference

Exact upper and lower bounds on the difference between the arithmetic and geometric means are obtained. The inequalities providing these bounds may be viewed, respectively, as a reverse Jensen inequality and an improvement of the direct Jensen inequality, in the case when the convex function is the exponential.

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