scholarly journals Growth Rates of Groups associated with Face 2-Coloured Triangulations and Directed Eulerian Digraphs on the Sphere

10.37236/5410 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Thomas A. McCourt
Keyword(s):  
Abelian Group ◽  
Lower Bounds ◽  
Growth Rates ◽  
Piecewise Linear ◽  
Upper And Lower Bounds ◽  
Torsion Subgroup ◽  
Pile Groups ◽  
Sand Pile ◽  
Black And White ◽  
Vertex Set

Let $\mathcal{G}$ be a properly face $2$-coloured (say black and white) piecewise-linear triangulation of the sphere with vertex set $V$. Consider the abelian group $\mathcal{A}_W$ generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $\mathcal{A}_B$ can be defined similarly, using black triangles. These groups are related in the following manner $\mathcal{A}_W\cong\mathcal{A}_B\cong\mathbb{Z}\oplus\mathbb{Z}\oplus\mathcal{C}$ where $\mathcal{C}$ is a finite abelian group.The finite torsion subgroup $\mathcal{C}$ is referred to as the canonical group of the triangulation. Let $m_t$ be the maximal order of $\mathcal{C}$ over all properly face 2-coloured spherical triangulations with $t$ triangles of each colour. By relating such a triangulation to certain directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to the triangulation's canonical group we provide improved upper and lower bounds for $\lim \sup_{t\rightarrow\infty}(m_t)^{1/t}$.

The number of sumsets in Abelian group

2020 ◽  
Vol 30 (5) ◽  
pp. 339-345
Author(s):  
Aleksandr A. Sapozhenko ◽  
Vahe G. Sargsyan
Keyword(s):  
Abelian Group ◽  
Lower Bounds ◽  
Upper And Lower Bounds

AbstractAsymptotic upper and lower bounds for the numbers of distinct subsets A + B in Abelian group of order n are derived, where |A|, |B| ≥ n(log n)−1/8.

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.

scholarly journals On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

2020 ◽  
Author(s):  
Stefan Kunis ◽  
Dominik Nagel
Keyword(s):  
Unit Circle ◽  
Lower Bounds ◽  
Condition Number ◽  
Growth Rates ◽  
Separation Distance ◽  
Upper And Lower Bounds ◽  
Spectral Condition ◽  
Sharp Constants ◽  
Vandermonde Matrices

Abstract We prove upper and lower bounds for the spectral condition number of rectangular Vandermonde matrices with nodes on the complex unit circle. The nodes are “off the grid,” pairs of nodes nearly collide, and the studied condition number grows linearly with the inverse separation distance. Such growth rates are known in greater generality if all nodes collide or for groups of colliding nodes. For pairs of nodes, we provide reasonable sharp constants that are independent of the number of nodes as long as non-colliding nodes are well-separated.

COVERING A SET OF POINTS WITH A MINIMUM NUMBER OF TURNS

2004 ◽  
Vol 14 (01n02) ◽  
pp. 105-114 ◽  
Author(s):  
MICHAEL J. COLLINS
Keyword(s):  
Euclidean Space ◽  
Lower Bounds ◽  
Piecewise Linear ◽  
Upper And Lower Bounds ◽  
Linear Path ◽  
Change Direction ◽  
Finite Set ◽  
Minimum Number ◽  
Pass Through ◽  
Set Of Points

Given a finite set of points in Euclidean space, we can ask what is the minimum number of times a piecewise-linear path must change direction in order to pass through all of them. We prove some new upper and lower bounds for the rectilinear version of this problem in which all motion is orthogonal to the coordinate axes. We also consider the more general case of arbitrary directions.

scholarly journals Antipodal graphs and digraphs

1993 ◽  
Vol 16 (3) ◽  
pp. 579-586 ◽  
Author(s):  
Garry Johns ◽  
Karen Sleno
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Vertex Set

The antipodal graph of a graphG, denoted byA(G), has the same vertex set asGwith an edge joining verticesuandvifd(u,v)is equal to the diameter ofG. (IfGis disconnected, thendiam G=∞.) This definition is extended to a digraphDwhere the arc(u,v)is included inA(D)ifd(u,v)is the diameter ofD. It is shown that a digraphDis an antipodal digraph if and only ifDis the antipodal digraph of its complement. This generalizes a known characterization for antipodal graphs and provides an improved proof. Examples and properties of antipodal digraphs are given. A digraphDis self-antipodal ifA(D)is isomorphic toD. Several characteristics of a self-antipodal digraphDare given including sharp upper and lower bounds on the size ofD. Similar results are given for self-antipodal graphs.

scholarly journals ON STRUCTURAL AND GRAPH THEORETIC PROPERTIES OF HIGHER ORDER DELAUNAY GRAPHS

2009 ◽  
Vol 19 (06) ◽  
pp. 595-615 ◽  
Author(s):  
MANUEL ABELLANAS ◽  
PROSENJIT BOSE ◽  
JESÚS GARCÍA ◽  
FERRAN HURTADO ◽  
CARLOS M. NICOLÁS ◽  
...  
Keyword(s):  
Cellular Networks ◽  
Lower Bounds ◽  
Higher Order ◽  
Combinatorial Structure ◽  
Upper And Lower Bounds ◽  
Coloring Problem ◽  
Graph Theoretic ◽  
Convex Position ◽  
Gabriel Graph ◽  
Vertex Set

Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-k Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the order-k Delaunay graph is connected for all k ≥ 0. We show that the order-k Gabriel graph, a subgraph of the order-k Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the order-k Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks.

Vertex-Neighbor-Scattering Number of Bipartite Graphs

2016 ◽  
Vol 27 (04) ◽  
pp. 501-509
Author(s):  
Zongtian Wei ◽  
Nannan Qi ◽  
Xiaokui Yue
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Bipartite Graphs ◽  
Upper And Lower Bounds ◽  
Number Of Components ◽  
Np Completeness ◽  
Vertex Set ◽  
Np Complete

Let G be a connected graph. A set of vertices [Formula: see text] is called subverted from G if each of the vertices in S and the neighbor of S in G are deleted from G. By G/S we denote the survival subgraph that remains after S is subverted from G. A vertex set S is called a cut-strategy of G if G/S is disconnected, a clique, or ø. The vertex-neighbor-scattering number of G is defined by [Formula: see text], where S is any cut-strategy of G, and ø(G/S) is the number of components of G/S. It is known that this parameter can be used to measure the vulnerability of spy networks and the computing problem of the parameter is NP-complete. In this paper, we discuss the vertex-neighbor-scattering number of bipartite graphs. The NP-completeness of the computing problem of this parameter is proven, and some upper and lower bounds of the parameter are also given.

The real genus spectrum of abelian groups

2019 ◽  
Vol 18 (08) ◽  
pp. 1950158
Author(s):  
Coy L. May ◽  
Jay Zimmerman
Keyword(s):  
Abelian Group ◽  
Lower Bounds ◽  
Necessary Conditions ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Upper And Lower Bounds ◽  
The Real ◽  
Odd Order ◽  
Positive Integers

Let [Formula: see text] denote the set of positive integers that may appear as the real genus of a finite abelian group. We obtain a set of (simple) necessary conditions for an integer [Formula: see text] to belong to [Formula: see text]. We also prove that the real genus of an abelian group is not congruent to 3 modulo 4 and that the genus of an abelian group of odd order is a multiple of 4. Finally, we obtain upper and lower bounds for the density of the set [Formula: see text].

scholarly journals Independent Transversals and Independent Coverings in Sparse Partite Graphs

1997 ◽  
Vol 6 (1) ◽  
pp. 115-125 ◽  
Author(s):  
RAPHAEL YUSTER
Keyword(s):  
Lower Bounds ◽  
Pairwise Disjoint ◽  
Independent Set ◽  
Independent Sets ◽  
Upper And Lower Bounds ◽  
Vertex Set

An [n, k, r]-partite graph is a graph whose vertex set, V, can be partitioned into n pairwise-disjoint independent sets, V1, …, Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 [les ] i < j [les ] n. An independent transversal in an [n, k, r]-partite graph is an independent set, T, consisting of n vertices, one from each Vi. An independent covering is a set of k pairwise-disjoint independent transversals. Let t(k, r) denote the maximal n for which every [n, k, r]-partite graph contains an independent transversal. Let c(k, r) be the maximal n for which every [n, k, r]-partite graph contains an independent covering. We give upper and lower bounds for these parameters. Furthermore, our bounds are constructive. These results improve and generalize previous results of Erdo″s, Gyárfás and Łuczak [5], for the case of graphs.

scholarly journals Two coloring problems on matrix graphs

2016 ◽  
Vol 08 (03) ◽  
pp. 1650053
Author(s):  
Zhe Han ◽  
Mei Lu
Keyword(s):  
Finite Field ◽  
Optical Networks ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Basic Properties ◽  
New Family ◽  
Minimum Number ◽  
Vertex Set ◽  
Positive Integers

In this paper, we propose a new family of graphs, matrix graphs, whose vertex set [Formula: see text] is the set of all [Formula: see text] matrices over a finite field [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text]. And any two matrices share an edge if the rank of their difference is [Formula: see text]. Next, we give some basic properties of such graphs and also consider two coloring problems on them. Let [Formula: see text] (resp., [Formula: see text]) denote the minimum number of colors necessary to color the above matrix graph so that no two vertices that are at a distance at most [Formula: see text] (resp., exactly [Formula: see text]) get the same color. These two problems were proposed in the study of scalability of optical networks. In this paper, we determine the exact value of [Formula: see text] and give some upper and lower bounds on [Formula: see text].

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