scholarly journals Jacobian Tori Associated with a Finite Graph and Its Abelian Covering Graphs

2000 ◽  
Vol 24 (2) ◽  
pp. 89-110 ◽  
Author(s):  
Motoko Kotani ◽  
Toshikazu Sunada
Keyword(s):  
Finite Graph ◽  
Abelian Covering

scholarly journals A VISIBILITY-BASED PURSUIT-EVASION PROBLEM

1999 ◽  
Vol 09 (04n05) ◽  
pp. 471-493 ◽  
Author(s):  
LEONIDAS J. GUIBAS ◽  
JEAN-CLAUDE LATOMBE ◽  
STEVEN M. LAVALLE ◽  
DAVID LIN ◽  
RAJEEV MOTWANI
Keyword(s):  
Finite Graph ◽  
Initial Position ◽  
Moving Target ◽  
Multiply Connected ◽  
Pursuit Evasion ◽  
Minimum Number ◽  
Evasion Problem ◽  
Free Spaces ◽  
Simply Connected ◽  
Motion Strategy

This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our study of this problem is motivated in part by robotics applications, such as surveillance with a mobile robot equipped with a camera that must find a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy for a single pursuer. For simply-connected free spaces, it is shown that the minimum number of pursuers required is Θ( lg  n). For multiply-connected free spaces, the bound is [Formula: see text] pursuers for a polygon that has n edges and h holes. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a finite graph that is constructed on the basis of critical information changes. It has been implemented and computed examples are shown.

scholarly journals Power-law bounds for critical long-range percolation below the upper-critical dimension

2021 ◽  
Author(s):  
Tom Hutchcroft
Keyword(s):  
Long Range ◽  
Power Law ◽  
Upper Bound ◽  
Mean Field ◽  
Maximum Size ◽  
Finite Graph ◽  
Critical Dimension ◽  
Point Function ◽  
Upper Critical Dimension ◽  
Infinite Cluster

AbstractWe study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

scholarly journals Two Short Proofs Concerning Tree-Decompositions

2002 ◽  
Vol 11 (6) ◽  
pp. 541-547 ◽  
Author(s):  
PATRICK BELLENBAUM ◽  
REINHARD DIESTEL
Keyword(s):  
Duality Theorem ◽  
Finite Graph ◽  
Tree Decomposition ◽  
Tree Width ◽  
Tree Decompositions

We give short proofs of the following two results: Thomas's theorem that every finite graph has a linked tree-decomposition of width no greater than its tree-width; and the ‘tree-width duality theorem’ of Seymour and Thomas, that the tree-width of a finite graph is exactly one less than the largest order of its brambles.

scholarly journals A Recursive Formula for the Reliability of a r-Uniform Complete Hypergraph and Its Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Ke Zhang ◽  
Haixing Zhao ◽  
Zhonglin Ye ◽  
Lixin Dong
Keyword(s):  
Recurrence Relations ◽  
Recursive Formula ◽  
Finite Graph ◽  
Hard Problem ◽  
Np Hard ◽  
Spanning Subgraph ◽  
Reliability Polynomial ◽  
Np Hard Problem

The reliability polynomial R(S,p) of a finite graph or hypergraph S=(V,E) gives the probability that the operational edges or hyperedges of S induce a connected spanning subgraph or subhypergraph, respectively, assuming that all (hyper)edges of S fail independently with an identical probability q=1-p. In this paper, we investigate the probability that the hyperedges of a hypergraph with randomly failing hyperedges induce a connected spanning subhypergraph. The computation of the reliability for (hyper)graphs is an NP-hard problem. We provide recurrence relations for the reliability of r-uniform complete hypergraphs with hyperedge failure. Consequently, we determine and calculate the number of connected spanning subhypergraphs with given size in the r-uniform complete hypergraphs.

scholarly journals Lower Bounds for Identifying Codes in some Infinite Grids

10.37236/394 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ryan Martin ◽  
Brendon Stanton
Keyword(s):  
Lower Bounds ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Locally Finite ◽  
Identifying Codes ◽  
Identifying Code ◽  
Hexagonal Grids ◽  
Closed Neighborhood ◽  
Definition Of ◽  
Infinite Grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

scholarly journals On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

2021 ◽  
pp. 1-13
Author(s):  
V. S. Guba
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Finite Graph ◽  
Vertex Degree ◽  
Doubling Property ◽  
Generating Set ◽  
Finite Set ◽  
Standard Set ◽  
Thompson’S Group ◽  
Average Vertex

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

Graphs with 6-Ways

1973 ◽  
Vol 25 (4) ◽  
pp. 687-692 ◽  
Author(s):  
John L. Leonard
Keyword(s):  
Finite Graph ◽  
Minimum Number ◽  
Multiple Edges

In a finite graph with no loops nor multiple edges, two points a and b are said to be connected by an r-way, or more explicitly, by a line r-way a — b if there are r paths, no two of which have lines in common (although they may share common points), which join a to b. In this note we demonstrate that any graph with n points and 3n — 2 or more lines must contain a pair of points joined by a 6-way, and that 3n — 2 is the minimum number of lines which guarantees the presence of a 6-way in a graph of n points.In the language of [3], this minimum number of lines needed to guarantee a 6-way is denoted U(n). For the background of this problem, the reader is referred to [3].

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