A lower bound on the average size of a connected vertex set of a graph

For a simple graph G with a vertex set V G and an edge set E G , a labeling f : V G ∪ E G ⟶ 1,2 , ⋯ , k is called a vertex irregular total k − labeling of G if for any two different vertices x and y in V G we have w t x ≠ w t y where w t x = f x + ∑ u ∈ V G f x u . The smallest positive integer k such that G has a vertex irregular total k − labeling is called the total vertex irregularity strength of G , denoted by tvs G . The lower bound of tvs G for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n ≥ 2 . Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is 3 n 2 + 1 / 2 .

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A New Characterization of Tree Medians with Applications to Distributed Algorithms

DAIMI Report Series ◽

10.7146/dpb.v20i364.6595 ◽

1991 ◽

Vol 20 (364) ◽

Cited By ~ 1

Author(s):

O. Gerstel ◽

Shmuel Zaks

Keyword(s):

Lower Bound ◽

Sorting Algorithm ◽

Message Complexity ◽

Ranking Problem ◽

Worst Case ◽

Sorting Problem ◽

Vertex Set ◽

Pass Through ◽

Case Number

A new characterization of tree medians is presented: we show that a vertex m is a median of a tree T with n vertices iff there exists a partition of the vertex set into [n/2] disjoint pairs (excluding m when n is odd), such that all the paths connecting the two vertices in any of the pairs pass through m. We show that in this case this sum is the largest possible among all such partitions, and we use this fact to discuss lower bounds on the message complexity of the distributed sorting problem. This lower bound implies that, given a network of a tree topology, choosing a median and then route all the information through it is the best possible strategy, in terms of worst-case number of messages sent during any execution of any distributed sorting algorithm. We also discuss the implications for networks of a general topology and for the distributed ranking problem.

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Alternative Branching Strategies in the Branch and Bound Algorithm by Using a K-Clique Covering Vertex Set for Maximum Clique Problems

International Journal of Quantitative Research and Modeling ◽

10.46336/ijqrm.v1i4.92 ◽

2020 ◽

Vol 1 (4) ◽

pp. 208-216

Author(s):

Mochamad Suyudi ◽

Asep K. Supriatna ◽

Sukono Sukono

Keyword(s):

Lower Bound ◽

Branch And Bound ◽

Maximum Clique ◽

Maximum Clique Problem ◽

Theory Problem ◽

Maximum Cardinality ◽

Arbitrary Graph ◽

Complete Subgraph ◽

Vertex Set ◽

The Difference

The maximum clique problem (MCP) is graph theory problem that demand complete subgraph with maximum cardinality (maximum clique) in arbitrary graph. Solving MCP usually use Branch and Bound (BnB) algorithm. In this paper, we will show how n + 1 color classes (where n is the difference between upper and lower bound) selected to form k-clique covering vertex set which later used for branching strategy can guarantee finding maximum clique.

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Bound Graph Polysemy

The Electronic Journal of Combinatorics ◽

10.37236/1521 ◽

2000 ◽

Vol 7 (1) ◽

Cited By ~ 2

Author(s):

Paul J. Tanenbaum

Keyword(s):

Lower Bound ◽

Partial Order ◽

Upper Bound ◽

Special Cases ◽

Vertex Set

Bound polysemy is the property of any pair $(G_1, G_2)$ of graphs on a shared vertex set $V$ for which there exists a partial order on $V$ such that any pair of vertices has an upper bound precisely when the pair is an edge in $G_1$ and a lower bound precisely when it is an edge in $G_2$. We examine several special cases and prove a characterization of the bound polysemic pairs that illuminates a connection with the squared graphs.

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Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.453 ◽

2009 ◽

Vol Vol. 11 no. 2 (Graph and Algorithms) ◽

Author(s):

Gábor Bacsó ◽

Zsolt Tuza

Keyword(s):

Lower Bound ◽

Polynomial Time ◽

Connected Graph ◽

Maximum Degree ◽

Free Graph ◽

Polynomial Time Algorithms ◽

Odd Cycle ◽

Vertex Set ◽

International Audience

Graphs and Algorithms International audience A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.

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An Empirical Estimation of the Underground Economy in Ghana

Economics Research International ◽

10.1155/2014/891237 ◽

2014 ◽

Vol 2014 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Edward Asiedu ◽

Thanasis Stengos

Keyword(s):

Lower Bound ◽

Underground Economy ◽

Parameter Estimates ◽

Empirical Estimation ◽

Demand Equation ◽

Long Run ◽

Currency Demand ◽

Currency Demand Approach ◽

Average Size

The main aim of this paper is to estimate the size of the underground economy in Ghana during the period 1983–2003. There is no agreement on the appropriate estimation approach to adopt to measure the size of the underground activities. To this end, we employ the well-applied currency demand approach in our measurement. Parameter estimates from the estimated currency demand equation are used in quantifying the ratio of “underground” to “measured” output/income for the Ghanaian economy. The estimated long-run average size of the underground economy to GDP for Ghana over the period is 40%. The underground economy is found to vary from a high of 54% in 1985 to a low of 25% in 1999. Estimates may represent lower bound estimates.

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On the Chromatic Number of Matching Kneser Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548319000178 ◽

2019 ◽

Vol 29 (1) ◽

pp. 1-21

Author(s):

Meysam Alishahi ◽

Hajiabolhassan Hossein

Keyword(s):

Lower Bound ◽

Chromatic Number ◽

Large Family ◽

Permutation Graphs ◽

Kneser Graph ◽

Ulam Theorem ◽

Turán Number ◽

Vertex Set ◽

Kneser Graphs ◽

Host Graph

AbstractIn an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.

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On differences of two squares

Open Mathematics ◽

10.1007/s11533-005-0007-0 ◽

2006 ◽

Vol 4 (1) ◽

pp. 110-122

Author(s):

Manfred Kühleitner ◽

Werner Nowak

Keyword(s):

Lower Bound ◽

Positive Integer ◽

Error Term ◽

Arithmetic Function ◽

Close Connection ◽

Mean Square ◽

Number Of Divisors ◽

Summatory Function ◽

Average Size ◽

Sharp Lower Bound

AbstractThe arithmetic function ρ(n) counts the number of ways to write a positive integer n as a difference of two squares. Its average size is described by the Dirichlet summatory function Σn≤x ρ(n), and in particular by the error term R(x) in the corresponding asymptotics. This article provides a sharp lower bound as well as two mean-square results for R(x), which illustrates the close connection between ρ(n) and the number-of-divisors function d(n).

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