The optimal pebbling of spindle graphs

Abstract Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The optimal pebbling number of G, denoted by πopt(G), is the smallest number m such that for some distribution of m pebbles on G, one pebble can be moved to any vertex of G by a sequence of pebbling moves. Let Pk be the path on k vertices. Snevily defined the n–k spindle graph as follows: take n copies of Pk and two extra vertices x and y, and then join the left endpoint (respectively, the right endpoint) of each Pk to x (respectively, y), the resulting graph is denoted by S(n, k), and called the n–k spindle graph. In this paper, we determine the optimal pebbling number for spindle graphs.

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Herscovici’s conjecture on C2n × G

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500718 ◽

2020 ◽

Vol 12 (05) ◽

pp. 2050071

Author(s):

A. Lourdusamy ◽

T. Mathivanan

Keyword(s):

Positive Integer ◽

Connected Graph ◽

Adjacent Vertex ◽

Connected Graphs ◽

Number Formula ◽

Pebbling Number

The [Formula: see text]-pebbling number, [Formula: see text], of a connected graph [Formula: see text], is the smallest positive integer such that from every placement of [Formula: see text] pebbles, [Formula: see text] pebbles can be moved to any specified target vertex by a sequence of pebbling moves, each move taking two pebbles off a vertex and placing one on an adjacent vertex. A graph [Formula: see text] satisfies the [Formula: see text]-pebbling property if [Formula: see text] pebbles can be moved to any specified vertex when the total starting number of pebbles is [Formula: see text], where [Formula: see text] is the number of vertices with at least one pebble. We show that the cycle [Formula: see text] satisfies the [Formula: see text]-pebbling property. Herscovici conjectured that for any connected graphs [Formula: see text] and [Formula: see text], [Formula: see text]. We prove Herscovici’s conjecture is true, when [Formula: see text] is an even cycle and for variety of graphs [Formula: see text] which satisfy the [Formula: see text]-pebbling property.

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The 2-pebbling property of squares of paths and Graham’s conjecture

Open Mathematics ◽

10.1515/math-2020-0009 ◽

2020 ◽

Vol 18 (1) ◽

pp. 87-92

Author(s):

Yueqing Li ◽

Yongsheng Ye

Keyword(s):

Connected Graph ◽

Adjacent Vertex ◽

Graham’S Conjecture ◽

Pebbling Number

Abstract A pebbling move on a graph G consists of taking two pebbles off one vertex and placing one pebble on an adjacent vertex. The pebbling number of a connected graph G, denoted by f(G), is the least n such that any distribution of n pebbles on G allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. In this paper, we determine the 2-pebbling property of squares of paths and Graham’s conjecture on $\begin{array}{} P_{2n}^2 \end{array} $.

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A note on pebbling bound for a product of Class 0 graphs

Asian-European Journal of Mathematics ◽

10.1142/s1793557122500310 ◽

2021 ◽

pp. 2250031

Author(s):

Nopparat Pleanmani ◽

Somnuek Worawiset

Keyword(s):

Upper Bound ◽

Connected Graph ◽

Adjacent Vertex ◽

Arbitrary Vertex ◽

Pebbling Number

Let [Formula: see text] be a connected graph. For a configuration of pebbles on the vertices of [Formula: see text], a pebbling move on [Formula: see text] is the process of taking two pebbles from a vertex and adding one of them on an adjacent vertex. The pebbling number of [Formula: see text], denoted by [Formula: see text], is the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. The graph [Formula: see text] is said to be of Class 0 if its pebbling number equals its order. For a Class [Formula: see text] connected graph [Formula: see text], we improve a recent upper bound for [Formula: see text] in terms of [Formula: see text].

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The 2-Pebbling Property of the Middle Graph of Fan Graphs

Journal of Applied Mathematics ◽

10.1155/2014/304514 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5

Author(s):

Yongsheng Ye ◽

Fang Liu ◽

Caixia Shi

Keyword(s):

Connected Graph ◽

Adjacent Vertex ◽

Arbitrary Vertex ◽

Pebbling Number ◽

Middle Graph

A pebbling move on a graphGconsists of taking two pebbles off one vertex and placing one pebble on an adjacent vertex. The pebbling number of a connected graphG, denoted byf(G), is the leastnsuch that any distribution ofnpebbles onGallows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. This paper determines the pebbling numbers and the 2-pebbling property of the middle graph of fan graphs.

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The optimal pebbling number of staircase graphs

Discrete Mathematics ◽

10.1016/j.disc.2018.06.042 ◽

2019 ◽

Vol 342 (7) ◽

pp. 2148-2157 ◽

Cited By ~ 1

Author(s):

Ervin Győri ◽

Gyula Y. Katona ◽

László F. Papp ◽

Casey Tompkins

Keyword(s):

Pebbling Number ◽

Optimal Pebbling

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Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383091950068x ◽

2019 ◽

Vol 11 (06) ◽

pp. 1950068

Author(s):

Nopparat Pleanmani

Keyword(s):

Bipartite Graph ◽

Network Optimization ◽

Connected Graph ◽

Complete Bipartite Graph ◽

Fixed Number ◽

Arbitrary Vertex ◽

Graph Pebbling ◽

Vertex Set ◽

Graham’S Conjecture ◽

Pebbling Number

A graph pebbling is a network optimization model for the transmission of consumable resources. A pebbling move on a connected graph [Formula: see text] is the process of removing two pebbles from a vertex and placing one of them on an adjacent vertex after configuration of a fixed number of pebbles on the vertex set of [Formula: see text]. The pebbling number of [Formula: see text], denoted by [Formula: see text], is defined to be the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. For connected graphs [Formula: see text] and [Formula: see text], Graham’s conjecture asserted that [Formula: see text]. In this paper, we show that such conjecture holds when [Formula: see text] is a complete bipartite graph with sufficiently large order in terms of [Formula: see text] and the order of [Formula: see text].

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On the Use of Entropy as a Measure of Dependence of Two Events

10.20944/preprints202106.0100.v2 ◽

2021 ◽

Author(s):

Valentin Iliev

Keyword(s):

Probability Space ◽

Entropy Function ◽

The Other ◽

Screening Tests ◽

Global Maximum ◽

Left Endpoint ◽

Positive Dependence ◽

The Right ◽

Measure Of Dependence ◽

The Impact

We define degree of dependence of two events A and B in a probability space by using Boltzmann-Shannon entropy function of an appropriate probability distribution produced by these events and depending on one parameter (the probability of intersection of A and B) varying within a closed interval I. The entropy function attains its global maximum when the events A and B are independent. The important particular case of discrete uniform probability space motivates this definition in the following way. The entropy function has a minimum at the left endpoint of I exactly when one of the events and the complement of the other are connected with the relation of inclusion (maximal negative dependence). It has a minimum at the right endpoint of I exactly when one of these events is included in the other (maximal positive dependence). Moreover, the deviation of the entropy from its maximum is equal to average information that carries one of the binary trials defined by A and B with respect to the other. As a consequence, the degree of dependence of A and B can be expressed in terms of information theory and is invariant with respect to the choice of unit of information. Using this formalism, we describe completely the screening tests and their reliability, measure efficacy of a vaccination, the impact of some events from the financial markets to other events, etc.

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GLOBAL CONTROLLABILITY OF A NONLINEAR KORTEWEG–DE VRIES EQUATION

Communications in Contemporary Mathematics ◽

10.1142/s0219199709003454 ◽

2009 ◽

Vol 11 (03) ◽

pp. 495-521 ◽

Cited By ~ 10

Author(s):

MARIANNE CHAPOULY

Keyword(s):

Boundary Control ◽

Vries Equation ◽

Boundary Value ◽

Exact Controllability ◽

Left Endpoint ◽

Bounded Interval ◽

First Derivative ◽

De Vries ◽

Positive Time ◽

The Right

We are interested in both the global exact controllability to the trajectories and in the global exact controllability of a nonlinear Korteweg–de Vries equation in a bounded interval. The local exact controllability to the trajectories by means of one boundary control, namely the boundary value at the left endpoint, has already been proved independently by Rosier, and Glass and Guerrero. We first introduce here two more controls: the boundary value at the right endpoint and the right member of the equation, assumed to be x-independent. Then, we prove that, thanks to these three controls, one has the global exact controllability to the trajectories, for any positive time T. Finally, we introduce a fourth control on the first derivative at the right endpoint, and we get the global exact controllability, for any positive time T.

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