scholarly journals Removing Symmetry in Circulant Graphs and Point-Block Incidence Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 166
Author(s):  
Josephine Brooks ◽  
Alvaro Carbonero ◽  
Joseph Vargas ◽  
Rigoberto Flórez ◽  
Brendan Rooney ◽  
...  
Keyword(s):  
Graph Theory ◽  
Design Theory ◽  
Difficult Problem ◽  
Combinatorial Design ◽  
Surprising Result ◽  
Cyclic Structure ◽  
Circulant Graphs ◽  
Minimum Number ◽  
Automorphism Of A Graph

An automorphism of a graph is a mapping of the vertices onto themselves such that connections between respective edges are preserved. A vertex v in a graph G is fixed if it is mapped to itself under every automorphism of G. The fixing number of a graph G is the minimum number of vertices, when fixed, fixes all of the vertices in G. The determination of fixing numbers is important as it can be useful in determining the group of automorphisms of a graph-a famous and difficult problem. Fixing numbers were introduced and initially studied by Gibbons and Laison, Erwin and Harary and Boutin. In this paper, we investigate fixing numbers for graphs with an underlying cyclic structure, which provides an inherent presence of symmetry. We first determine fixing numbers for circulant graphs, showing in many cases the fixing number is 2. However, we also show that circulant graphs with twins, which are pairs of vertices with the same neighbourhoods, have considerably higher fixing numbers. This is the first paper that investigates fixing numbers of point-block incidence graphs, which lie at the intersection of graph theory and combinatorial design theory. We also present a surprising result-identifying infinite families of graphs in which fixing any vertex fixes every vertex, thus removing all symmetries from the graph.

scholarly journals Butterfly Triple System Algorithm Based on Graph Theory

2021 ◽  
Vol 21 (No.1) ◽  
pp. 27-49
Author(s):  
Raja'i Mohammad Aldiabat ◽  
Haslinda Ibrahim ◽  
Sharmila Karim
Keyword(s):  
Graph Theory ◽  
Special Reference ◽  
Complete Graph ◽  
Triple System ◽  
Design Theory ◽  
Combinatorial Design ◽  
Algorithm Development ◽  
Cycle System ◽  
New Type ◽  
Insight Into

In combinatorial design theory, clustering elements into a set of three elements is the heart of classifying data. This article will provide insight into formulating algorithm for a new type of triple system, called a Butterfly triple system. Basically, in this algorithm development, a starter of cyclic near-resolvable ((v-1)/2)-cycle system of the 2-fold complete graph 2K_v is employed to construct the starter of cyclic ((v-1)/2)-star decomposition of 2K_v. These starters were then decomposed into triples and classified as a starter of a cyclic Butterfly triple system. The obtained starter set generated a triple system of order A special reference for case 𝑣𝑣 ≡ 9 (mod 12) was presented to demonstrate the development of the Butterfly triple system.

Graph Theory and Probability

1959 ◽  
Vol 11 ◽  
pp. 34-38 ◽  
Author(s):  
P. Erdös
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Complete Graph ◽  
Explicit Construction ◽  
Difficult Problem ◽  
Complete Subgraph ◽  
Image Position ◽  
Set Of Points

A well-known theorem of Ramsay (8; 9) states that to every n there exists a smallest integer g(n) so that every graph of g(n) vertices contains either a set of n independent points or a complete graph of order n, but there exists a graph of g(n) — 1 vertices which does not contain a complete subgraph of n vertices and also does not contain a set of n independent points. (A graph is called complete if every two of its vertices are connected by an edge; a set of points is called independent if no two of its points are connected by an edge.) The determination of g(n) seems a very difficult problem; the best inequalities for g(n) are (3)It is not even known that g(n)1/n tends to a limit. The lower bound in (1) has been obtained by combinatorial and probabilistic arguments without an explicit construction.

Determination of the minimum number of technological bases of a part

2020 ◽  
pp. 73-75
Author(s):  
B.M. Bazrov ◽  
T.M. Gaynutdinov
Keyword(s):  
Cost Price ◽  
Labor Input ◽  
Part Surface ◽  
Technological Base ◽  
Size Accuracy ◽  
Minimum Number ◽  
The Cost ◽  
Input Cost ◽  
Selection Of

The selection of technological bases is considered before the choice of the type of billet and the development of the route of the technological process. A technique is proposed for selecting the minimum number of sets of technological bases according to the criterion of equality in the cost price of manufacturing the part according to the principle of unity and combination of bases at this stage. Keywords: part, surface, coordinating size, accuracy, design and technological base, labor input, cost price. [email protected]

Tribology of rail bogie

2018 ◽  
Vol 77 (4) ◽  
pp. 230-240
Author(s):  
D. P. Markov
Keyword(s):  
Contact Fatigue ◽  
Basic Element ◽  
Difficult Problem ◽  
Railway Track ◽  
Rolling Stock ◽  
Kinematic Parameters ◽  
Approximate Estimation ◽  
Railway Bogie ◽  
Classical Calculation

Railway bogie is the basic element that determines the force, kinematic, power and other parameters of the rolling stock, and its movement in the railway track has not been studied enough. Classical calculation of the kinematic and dynamic parameters of the bogie's motion with the determination of the position of its center of rotation, the instantaneous axes of rotation of wheelsets, the magnitudes and directions of all forces present a difficult problem even in quasi-static theory. The paper shows a simplified method that allows one to explain, within the limits of one article, the main kinematic and force parameters of the bogie movement (installation angles, clearance between the wheel flanges and side surfaces of the rails), wear and contact damage to the wheels and rails. Tribology of the railway bogie is an important part of transport tribology, the foundation of the theory of wheel-rail tribosystem, without which it is impossible to understand the mechanisms of catastrophic wear, derailments, contact fatigue, cohesion of wheels and rails. In the article basic questions are considered, without which it is impossible to analyze the movement of the bogie: physical foundations of wheel movement along the rail, types of relative motion of contacting bodies, tribological characteristics linking the force and kinematic parameters of the bogie. Kinematics and dynamics of a two-wheeled bogie-rail bicycle are analyzed instead of a single wheel and a wheelset, which makes it clearer and easier to explain how and what forces act on the bogie and how they affect on its position in the rail track. To calculate the motion parameters of a four-wheeled bogie, it is represented as two two-wheeled, moving each on its own rail. Connections between them are replaced by moments with respect to the point of contact between the flange of the guide wheel and the rail. This approach made it possible to give an approximate estimation of the main kinematic and force parameters of the motion of an ideal bogie (without axes skewing) in curves, to understand how the corners of the bogie installation and the gaps between the flanges of the wheels and rails vary when moving with different speeds, how wear and contact injuries arise and to give recommendations for their assessment and elimination.

scholarly journals Breakout Group Allocation Schedules and the Social Golfer Problem with Adjacent Group Sizes

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 13
Author(s):  
Alice Miller ◽  
Matthew Barr ◽  
William Kavanagh ◽  
Ivaylo Valkov ◽  
Helen C. Purchase
Keyword(s):  
Design Theory ◽  
Teaching Experience ◽  
Combinatorial Design ◽  
Breakout Group ◽  
Online Resource ◽  
The Social ◽  
Adjacent Group ◽  
Group Allocation ◽  
Novel Variation ◽  
The Ideal

The current pandemic has led schools and universities to turn to online meeting software solutions such as Zoom and Microsoft Teams. The teaching experience can be enhanced via the use of breakout rooms for small group interaction. Over the course of a class (or over several classes), the class will be allocated to breakout groups multiple times over several rounds. It is desirable to mix the groups as much as possible, the ideal being that no two students appear in the same group in more than one round. In this paper, we discuss how the problem of scheduling balanced allocations of students to sequential breakout rooms directly corresponds to a novel variation of a well-known problem in combinatorics (the social golfer problem), which we call the social golfer problem with adjacent group sizes. We explain how solutions to this problem can be obtained using constructions from combinatorial design theory and how they can be used to obtain good, balanced breakout room allocation schedules. We present our solutions for up to 50 students and introduce an online resource that educators can access to immediately generate suitable allocation schedules.

A Hybrid Approach to Computer-Aided Process Planning for Prismatic Parts

1994 ◽  
Author(s):  
Y. F. Zhang ◽  
A. Y. C. Nee ◽  
J. Y. H. Fuh
Keyword(s):  
Process Planning ◽  
Hybrid Approach ◽  
Optimization Method ◽  
Optimal Operation ◽  
Operation Sequencing ◽  
Computer Aided Process Planning ◽  
Prismatic Parts ◽  
Minimum Number ◽  
Set Up

Abstract One of the most difficult tasks in automated process planning is the determination of operation sequencing. This paper describes a hybrid approach for identifying the optimal operation sequence of machining prismatic parts on a three-axis milling machining centre. In the proposed methodology, the operation sequencing is carried out in two levels of planning: set-up planning and operation planning. Various constraints on the precedence relationships between features are identified and rules and heuristics are created. Based on the precedence relationships between features, an optimization method is developed to find the optimal plan(s) with minimum number of set-ups in which the conflict between the feature precedence relationships and set-up sequence is avoided. For each set-up, an optimal feature machining sequence with minimum number of tool changes is also determined using a developed algorithm. The proposed system is still under development and the hybrid approach is partially implemented. An example is provided to demonstrate this approach.

A Simple Testing Procedure for near Infrared Instruments

1998 ◽  
Vol 6 (A) ◽  
pp. A163-A170 ◽  
Author(s):  
B. Barabás
Keyword(s):  
Error Compensation ◽  
Near Infrared ◽  
Testing Procedure ◽  
Wheat Protein ◽  
Calibration Equation ◽  
Minimum Number ◽  
Mean Square Difference ◽  
Infrared Instruments ◽  
Protein Measurement

The testing and adjusting procedure of near infrared (NIR) spectrophotometers is based on the measurement of some standards and, if necessary, on the adjustment of the constants in the calibration equation. For this work some use few standards, whereas others use 20 or more. This work was aimed to determine the range of error compensation and the minimum number of standards required. The experiments were applied to wheat protein measurement using two scanning spectrophotometers. The errors in the NIR measurements were characterised as bias, skew, error derived from skew ( Eskew) and standard error of difference corrected for bias and skew ( SEDc) parameters and supposed that errors derived from the change in the wavelength or reflectance of the instrument. The confidence intervals of bias and skew, derived from duplicate measurements of various numbers of wheat standards, were used to determine the minimum number of standards required. The range of error compensation was defined with those bias values, where SEDc was smaller, than an acceptable limit. The range of compensation corresponded to a bias value of ± 8 g kg−1 for wheat protein measurements. The detection of error of measurements required 4 wheat standards. The elimination of errors of bias and skew required 9 standards within the above limits. The developed procedure was tested in case of real instrument error. Diminishing a bias from 5.2 g kg−1 to 0.7 g kg−1 and the root mean square difference ( RMSD) to an acceptable level required the use of 9 standards, similar to the model experiment. The simplicity and rapidity (about 10 min) of the procedure enabled the routine test of NIR instruments. The range of error compensation and the number of standards referred to wheat protein. The simple modelling procedure proved also suitable for the determination of these values for other components and under other measuring conditions.

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