scholarly journals Mixed Moore Cayley Graphs

2017 ◽  
Vol 17 (03n04) ◽  
pp. 1741010
Author(s):  
GRAHAME ERSKINE
Keyword(s):  
Infinite Number ◽  
Upper Bound ◽  
Maximum Degree ◽  
Cayley Graphs ◽  
Computational Techniques ◽  
Elementary Group ◽  
Recent Interest ◽  
Mixed Graphs ◽  
Rule Out

The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree r and directed out-degree z, a straightforward counting argument yields an upper bound M(z, r, 2) = (z+r)2+z+1 for the order of the graph. Apart from the case r = 1, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there are an infinite number of feasible pairs (r, z) where the existence of mixed Moore graphs with these parameters is unknown. We use a combination of elementary group-theoretical arguments and computational techniques to rule out the existence of further examples of mixed Cayley graphs attaining the Moore bound for all orders up to 485.

scholarly journals On the Strong Equitable Vertex 2-Arboricity of Complete Bipartite Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1778
Author(s):  
Fangyun Tao ◽  
Ting Jin ◽  
Yiyou Tu
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Equitable Partition ◽  
Special Cases ◽  
Vertex Set ◽  
Complete Bipartite Graphs

An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by vak≡(G), is the smallest integer t such that G can be equitably partitioned into t′ induced forests for every t′≥t, where the maximum degree of each induced forest is at most k. In this paper, we provide a general upper bound for va2≡(Kn,n). Exact values are obtained in some special cases.

The Rates of Growth of the Galton–Watson Process in Varying Environments

1994 ◽  
Vol 26 (3) ◽  
pp. 698-714 ◽  
Author(s):  
J. C. D'Souza
Keyword(s):  
Infinite Number ◽  
Upper Bound ◽  
Sufficient Condition ◽  
Rate Of Growth ◽  
Varying Environments ◽  
Watson Process ◽  
Rates Of Growth

Let {Zn} be a supercritical Galton–Watson process in varying environments, and W be the limit of the non-negative martingale {Zn/EZn}. Under a condition which ensures that W is not identically equal to zero we give an upper bound on the possible rates of growth of the process on the set {W = 0}, and find a sufficient condition for the process to have only one rate of growth. We also give an example of a process whose offspring distributions have bounded pth moments, for some p > 1, and which has an infinite number of rates of growth.

A Simplified Kinematic Method for 3D Limit Analysis

2016 ◽  
Vol 846 ◽  
pp. 342-347 ◽  
Author(s):  
J.P. Hambleton ◽  
Scott William Sloan
Keyword(s):  
Upper Bound ◽  
Limit Analysis ◽  
Three Dimensional ◽  
Yield Condition ◽  
Limit Load ◽  
Computational Techniques ◽  
Simple Formulation ◽  
Planar Velocity ◽  
Collapse Mechanisms ◽  
Second Order Cone

The kinematic (upper bound) method of limit analysis is a powerful technique for evaluating rigorous bounds on limit loads that are often very close to the true limit load. While generalized computational techniques for two-dimensional (e.g., plane strain) problems are well established, methods applicable to three-dimensional problems are relatively underdeveloped and underutilized, due in large part to the cumbersome nature of the calculations for analytical solutions and the large computation times required for numerical approaches. This paper proposes a simple formulation for three-dimensional limit analysis that considers material obeying the Mohr-Coulomb yield condition and collapse mechanisms consisting of sliding rigid blocks separated by planar velocity discontinuities. A key advantage of the approach is its reliance on a minimal number of unknowns, can dramatically reduce processing time. The paper focuses specifically on tetrahedral blocks, although extension to alternative geometries is straightforward. For an arbitrary but fixed arrangement of blocks, the procedure for computing the unknown block velocities that yield the least upper bound is expressed as a second-order cone programming problem that can be easily solved using widely available optimization codes. The paper concludes with a simple example and remarks regarding extensions of the work.

scholarly journals An improved upper bound for the order of mixed graphs

2018 ◽  
Vol 341 (10) ◽  
pp. 2872-2877 ◽  
Author(s):  
C. Dalfó ◽  
M.A. Fiol ◽  
N. López
Keyword(s):  
Upper Bound ◽  
Mixed Graphs

scholarly journals Integer index in trees of diameter 4

Filomat ◽  
2014 ◽  
Vol 28 (2) ◽  
pp. 241-248 ◽  
Author(s):  
Laura Patuzzi ◽  
Freitas de ◽  
Renata Del-Vecchio
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Infinite Family

We characterize when a tree of diameter 4 has integer index and we provide examples of infinite families of non-integral trees with integer index. We also determine a tight upper bound for the index of any tree of diameter 4 based on its maximum degree. Moreover, we present a new infinite family of integral trees of diameter 4.

scholarly journals Strong rainbow connection numbers of toroidal meshes

2018 ◽  
Vol 10 (03) ◽  
pp. 1850039
Author(s):  
Yulong Wei ◽  
Min Xu ◽  
Kaishun Wang
Keyword(s):  
Upper Bound ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Negative Answer ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection

In 2011, Li et al. [The (strong) rainbow connection numbers of Cayley graphs on Abelian groups, Comput. Math. Appl. 62(11) (2011) 4082–4088] obtained an upper bound of the strong rainbow connection number of an [Formula: see text]-dimensional undirected toroidal mesh. In this paper, this bound is improved. As a result, we give a negative answer to their problem.

scholarly journals Sparse Highly Connected Spanning Subgraphs in Dense Directed Graphs

2018 ◽  
Vol 28 (3) ◽  
pp. 423-464 ◽  
Author(s):  
DONG YEAP KANG
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Analogous Result ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Additional Term ◽  
Spanning Subgraphs ◽  
Spanning Subgraph ◽  
Polynomial Time Algorithms ◽  
Bipartite Digraph

Mader proved that every strongly k-connected n-vertex digraph contains a strongly k-connected spanning subgraph with at most 2kn - 2k2 edges, where equality holds for the complete bipartite digraph DKk,n-k. For dense strongly k-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly k-connected n-vertex digraph D contains a strongly k-connected spanning subgraph with at most kn + 800k(k + Δ(D)) edges, where Δ(D) denotes the maximum degree of the complement of the underlying undirected graph of a digraph D. Here, the additional term 800k(k + Δ(D)) is tight up to multiplicative and additive constants. As a corollary, this implies that every strongly k-connected n-vertex semicomplete digraph contains a strongly k-connected spanning subgraph with at most kn + 800k2 edges, which is essentially optimal since 800k2 cannot be reduced to the number less than k(k - 1)/2.We also prove an analogous result for strongly k-arc-connected directed multigraphs. Both proofs yield polynomial-time algorithms.

An Upper Bound on the Least Inert Prime in a Real Quadratic Field

2000 ◽  
Vol 52 (2) ◽  
pp. 369-380 ◽  
Author(s):  
Andrew Granville ◽  
R. A. Mollin ◽  
H. C. Williams
Keyword(s):  
Upper Bound ◽  
Quadratic Field ◽  
Computational Techniques ◽  
Real Quadratic Field ◽  
Kronecker Symbol ◽  
Fundamental Discriminant ◽  
Real Quadratic

AbstractIt is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D > 3705, there is always at least one prime p < √D/2 such that the Kronecker symbol (D/p) = −1.

scholarly journals Cosmic microwave background constraints on a physical model of reionization

2020 ◽  
Vol 501 (1) ◽  
pp. L7-L11
Author(s):  
Tirthankar Roy Choudhury ◽  
Suvodip Mukherjee ◽  
Sourabh Paul
Keyword(s):  
Cosmic Microwave Background ◽  
Upper Bound ◽  
Minimum Mass ◽  
Microwave Background ◽  
Mode Amplitude ◽  
Primordial Gravitational Waves ◽  
Model Predictions ◽  
Free Parameters ◽  
Significant Region ◽  
Rule Out

ABSTRACT We study constraints on allowed reionization histories by comparing predictions of a physical seminumerical model with secondary temperature and polarization anisotropies of the cosmic microwave background (CMB). Our model has four free parameters characterizing the evolution of ionizing efficiency ζ and the minimum mass Mmin of haloes that can produce ionizing radiation. Comparing the model predictions with the presently available data of the optical depth τ and kinematic Sunyaev–Zeldovich signal, we find that we can already rule out a significant region of the parameter space. We limit the duration of reionization $\Delta z= 1.30^{+0.19}_{-0.60}$ (Δz &lt; 2.9 at $99{{\ \rm per\ cent}}$ C.L.), one of the tightest constraints on the parameter. The constraints mildly favour Mmin ≳ 109 M⊙ (at $68{{\ \rm per\ cent}}$ C.L.) at z ∼ 8, thus indicating the presence of reionization feedback. Our analysis provides an upper bound on the secondary B-mode amplitude $D_{l=200}^{BB} \lt 18$ nK2 at $99{{\ \rm per\ cent}}$ C.L. We also study how the constraints can be further tightened with upcoming space- and ground-based CMB missions. Our study, which relies solely on CMB data, has implications not only for upcoming CMB surveys for detecting primordial gravitational waves but also redshifted 21 cm studies.

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