The Minimum Vertex Degree of a Graph on Uniform Points in [0, 1]d

1997 ◽  
Vol 29 (3) ◽  
pp. 582-594 ◽  
Author(s):  
Martin J. B. Appel ◽  
Ralph P. Russo
Keyword(s):  
Random Graph ◽  
Nearest Neighbor ◽  
Vertex Degree ◽  
Degree Zero ◽  
Asymptotic Results ◽  
Limiting Behavior ◽  
Edge Distance ◽  
Vertex Set ◽  
Image Position

This article continues an investigation begun in [2]. A random graph Gn(x) is constructed on independent random points U1, · ··, Un distributed uniformly on [0, 1]d, d ≧ 1, in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 < x < 1.Almost-sure asymptotic results are obtained for the convergence/divergence of the minimum vertex degree of the random graph, as the number n of points becomes large and the edge distance x is allowed to vary with n. The largest nearest neighbor link dn, the smallest x such that Gn(x) has no vertices of degree zero, is shown to satisfy Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be complete, a.s. These criteria imply a.s. limiting behavior of the diameter of the vertex set.

The Minimum Vertex Degree of a Graph on Uniform Points in [0, 1]d

1997 ◽  
Vol 29 (03) ◽  
pp. 582-594
Author(s):  
Martin J. B. Appel ◽  
Ralph P. Russo
Keyword(s):  
Random Graph ◽  
Nearest Neighbor ◽  
Vertex Degree ◽  
Degree Zero ◽  
Asymptotic Results ◽  
Limiting Behavior ◽  
Edge Distance ◽  
Vertex Set ◽  
Image Position

This article continues an investigation begun in [2]. A random graph Gn (x) is constructed on independent random points U 1, · ··, Un distributed uniformly on [0, 1] d , d ≧ 1, in which two distinct such points are joined by an edge if the l ∞-distance between them is at most some prescribed value 0 &lt; x &lt; 1. Almost-sure asymptotic results are obtained for the convergence/divergence of the minimum vertex degree of the random graph, as the number n of points becomes large and the edge distance x is allowed to vary with n. The largest nearest neighbor link dn, the smallest x such that Gn (x) has no vertices of degree zero, is shown to satisfy Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be complete, a.s. These criteria imply a.s. limiting behavior of the diameter of the vertex set.

On colouring random graphs

1975 ◽  
Vol 77 (2) ◽  
pp. 313-324 ◽  
Author(s):  
G. R. Grimmett ◽  
C. J. H. McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Subgraph ◽  
Vertex Set ◽  
Image Position

AbstractLet ωn denote a random graph with vertex set {1, 2, …, n}, such that each edge is present with a prescribed probability p, independently of the presence or absence of any other edges. We show that the number of vertices in the largest complete subgraph of ωn is, with probability one,

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]d

1997 ◽  
Vol 29 (03) ◽  
pp. 567-581 ◽  
Author(s):  
Martin J. B. Appel ◽  
Ralph P. Russo
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Independence Number ◽  
Rates Of Convergence ◽  
Clique Number ◽  
Vertex Degree ◽  
Edge Distance

On independent random points U 1 ,· ··,Un distributed uniformly on [0, 1] d , a random graph Gn (x) is constructed in which two distinct such points are joined by an edge if the l ∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn } are provided which guarantee the random graph to be empty of edges, a.s.

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]d

1997 ◽  
Vol 29 (3) ◽  
pp. 567-581 ◽  
Author(s):  
Martin J. B. Appel ◽  
Ralph P. Russo
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Independence Number ◽  
Rates Of Convergence ◽  
Clique Number ◽  
Vertex Degree ◽  
Edge Distance

On independent random points U1,· ··,Un distributed uniformly on [0, 1]d, a random graph Gn(x) is constructed in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be empty of edges, a.s.

scholarly journals The Distribution of Heights of Binary Trees and Other Simple Trees

1993 ◽  
Vol 2 (2) ◽  
pp. 145-156 ◽  
Author(s):  
Philippe Flajolet ◽  
Zhicheng Gao ◽  
Andrew Odlyzko ◽  
Bruce Richmond
Keyword(s):  
Asymptotic Formula ◽  
Positive Constant ◽  
Binary Trees ◽  
Asymptotic Results ◽  
Image Position

The number, , of rooted plane binary trees of height ≤ h with n internal nodes is shown to satisfyuniformly for δ−1(log n)−1/2 ≤ β ≤ δ(log n)1/2, where and δ is a positive constant. An asymptotic formula for is derived for h = cn, where 0 < c < 1. Bounds for are also derived for large and small heights. The methods apply to any simple family of trees, and the general asymptotic results are stated.

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.

Weighted Norm Inequalities for Fractional Integral Operators With Rough Kernel

1998 ◽  
Vol 50 (1) ◽  
pp. 29-39 ◽  
Author(s):  
Yong Ding ◽  
Shanzhen Lu
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Integral Operators ◽  
Rough Kernel ◽  
Degree Zero ◽  
Weighted Norm ◽  
Weighted Norm Inequalities ◽  
Norm Inequalities ◽  
Fractional Integral Operators ◽  
Image Position

AbstractGiven function Ω on ℝn , we define the fractional maximal operator and the fractional integral operator by and respectively, where 0 < α < n. In this paper we study the weighted norm inequalities of MΩα and TΩα for appropriate α, s and A(p, q) weights in the case that Ω∈ Ls(Sn-1)(s> 1), homogeneous of degree zero.

scholarly journals On Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

On random mappings with a single attracting centre

1987 ◽  
Vol 24 (1) ◽  
pp. 258-264 ◽  
Author(s):  
Ljuben R. Mutafchiev
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Vector ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Random Mappings ◽  
Vertex Set ◽  
Independent Identically Distributed ◽  
Independent And Identically Distributed

We consider the random vector T = (T(0), ···, T(n)) with independent identically distributed coordinates such that Pr{T(i) = j} = Pj, j = 0, 1, ···, n, Σ . A realization of T can be viewed as a random graph GT with vertices {0, ···, n} and arcs {(0, T(0)), ···, (n, T(n))}. For each T we partition the vertex-set of GT into three disjoint groups and study the joint probability distribution of their cardinalities. Assuming that we observe the asymptotics of this distribution, as n → ∞, for all possible values of P0. It turns out that in some cases these cardinalities are asymptotically independent and identically distributed.

Graphs with a locally linear group of automorphisms

1995 ◽  
Vol 118 (2) ◽  
pp. 191-206 ◽  
Author(s):  
V. I. Trofimov ◽  
R. M. Weiss
Keyword(s):  
Permutation Group ◽  
Linear Group ◽  
Undirected Graph ◽  
Group Of Automorphisms ◽  
Vertex Set ◽  
Image Position ◽  
Pointwise Stabilizer ◽  
Locally Linear

Let Γ be an undirected graph, V(Γ) the vertex set of Γ and G a subgroup of aut(Γ). For each vertex x ↦ V(Γ), let Γx denote the set of vertices adjacent to x in Γ and the permutation group induced on Γx. by the stabilizer Gx. For each i ≥ 1, will denote the pointwise stabilizer in Gx of the set of vertices at distance at most i from x in Γ. Letfor each i ≥ 1 and any set of vertices x, y, …, z of Γ. An s-path (or s-arc) is an (s + 1)-tuple (x0, x1, … xs) of vertices such that xi ↦ Γxi–1 for 1 ≤ i ≤ s and xi ╪ xi–2 for 2 ≤ i ≤ s.

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