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### Large circulant graphs of fixed diameter and arbitrary degree [article]

David Bevan, Grahame Erskine, Robert Lewis
2017 arXiv   pre-print
To date, attempts to generate families of large circulant graphs of arbitrary degree for a given diameter have concentrated mainly on the diameter 2 case.  ...  We also present a revised table of largest known circulant graphs of small degree and diameter.  ...  We also construct large directed circulant graphs of diameters k = 2, . . . , 9 and arbitrary large degree.  ...

### Large circulant graphs of fixed diameter and arbitrary degree

David Bevan, Grahame Erskine, Robert Lewis
2017 Ars Mathematica Contemporanea
To date, attempts to generate families of large circulant graphs of arbitrary degree for a given diameter have concentrated mainly on the diameter 2 case.  ...  We also present a revised table of largest known circulant graphs of small degree and diameter.  ...  We also construct large directed circulant graphs of diameters k = 2, . . . , 9 and arbitrary large degree.  ...

### The Degree-Diameter Problem for Circulant Graphs of Degree 8 and 9 [article]

Robert Lewis
2014 arXiv   pre-print
This paper considers the degree-diameter problem for undirected circulant graphs. The focus is on extremal graphs of given (small) degree and arbitrary diameter.  ...  Candidate graphs are defined as functions of the diameter for both degree 8 and degree 9. They are proven to be extremal for small diameters.  ...  For circulant graphs there are two main areas of research: largest graphs of given (small) degree and arbitrary diameter, and of given (small) diameter and arbitrary degree.  ...

### The Degree-Diameter Problem for Circulant Graphs of Degree 8 and 9

Robert R. Lewis
2014 Electronic Journal of Combinatorics
This paper considers the degree-diameter problem for undirected circulant graphs. The focus is on extremal graphs of given (small) degree and arbitrary diameter.  ...  Candidate graphs are defined as functions of the diameter for both degree 8 and degree 9. They are proven to be extremal for small diameters.  ...  Acknowledgements I would like to thank Professor JozefŠiráň for reviewing the draft and making many helpful comments to improve the presentation.  ...

### Page 6541 of Mathematical Reviews Vol. , Issue 93m [page]

1993 Mathematical Reviews
An analogous lower bound on the diameter of families of circulant graphs of fixed degree and given Z-rank of the generators is given.”  ...  For fixed degree k and large p, we obtain a lower bound of order @(p'/?\*)), where g is Euler’s totient function.  ...

### Algebraic and computer-based methods in the undirected degree/diameter problem - A brief survey English

Hebert Perez-Roses
2014 Electronic Journal of Graph Theory and Applications
Problem 1 (Degree/Diameter problem for undirected graphs). Given positive integers ∆ and D, find the largest possible number of vertices N ∆,D of a graph of maximum degreeand diameter D.  ...  This paper discusses the most popular algebraic techniques and computational methods that have been used to construct large undirected graphs with given degree and diameter.  ...  Now let N circ ∆,D be the number of vertices of the largest circulant graph with degreeand diameter D.  ...

### Improved upper bounds for the order of some classes of Abelian Cayley and circulant graphs of diameter two [article]

Robert R. Lewis
2015 arXiv   pre-print
order of general diameter 2 circulant graphs of arbitrary degree.  ...  In the degree-diameter problem for Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d there is a wide gap between the best lower and upper bounds valid for all d, being quadratic  ...  This paper considers the degree-diameter problem for undirected Cayley graphs of diameter 2 and arbitrary degree, of cyclic groups and of Abelian groups in general.  ...

### Page 421 of Mathematical Reviews Vol. , Issue 95a [page]

1995 Mathematical Reviews
In Section 2, we give a regular MBD with n vertices and degree |log, | which is in the class of circulant digraphs.  ...  Summary: “We compute the exact fault diameter of the star graph.  ...

### Ergodic Effects in Token Circulation [chapter]

2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms
The remaining sections are devoted to a proof of the results, starting with an exposition of the main structural lemmas on mixing properties of circulations in Section 3 and 4.  ...  To prove its existence, we build on links between the RR dynamics and the structure of Eulerian circuits in the graph.  ...  Acknowledgments We thank Jurek Czyzowicz and Jukka Suomela for fruitful discussions. We also thank an anonymous referee for valuable comments.  ...

### Ergodic Effects in Token Circulation [article]

2017 arXiv   pre-print
Formally, for a system of k particles in a graph of m edges, during any interval of length T, this time-averaged value is k/m ±O(1/T), whenever (m,k) = O(1) (and so, e.g., whenever m is a prime number)  ...  These results are proved through sum set methods and are likely to be of independent interest.  ...  Acknowledgments We thank Jurek Czyzowicz and Jukka Suomela for fruitful discussions. We also thank an anonymous referee for valuable comments.  ...

### Research on Topology Structure Analysis of Several Interconnection Networks

Shu-na ZHOU
2016 DEStech Transactions on Computer Science and Engineering
In this paper, it is proved that the routing selection of some graphs can achieve the minimum fault tolerance.  ...  There are many indicators to evaluate the performance of an interconnection network, this paper introduces the basic methods and principles of an interconnection network, and analyzes the characteristics  ...  For a given graph, if fix its diameter and the maximum degree, then the maximum number of components that can be interconnected are controlled by a function of diameter and maximum degree.  ...

### On the defect of vertex-transitive graphs of given degree and diameter

Geoffrey Exoo, Robert Jajcay, Martin Mačaj, Jozef Širáň
2018 Journal of combinatorial theory. Series B (Print)
We consider the problem of finding largest vertex-transitive graphs of given degree and diameter.  ...  Using two classical number theory results due to Niven and Erdős, we prove that for any fixed degree ∆ ≥ 3 and any positive integer δ, the order of a largest vertex-transitive ∆-regular graph of diameter  ...  Acknowledgments We thank the referees for their useful corrections and suggestions that made us clarify several parts of the paper which ultimately led to better bounds.  ...

### Multiplicative circulant networks topological properties and communication algorithms

Ivan Stojmenović
1997 Discrete Applied Mathematics
We prove that for even r > 2 the diameter of the MC(r, k) network is kr/2 -jk/2] which is smaller than the diameter of the corresponding torus.  ...  We also determine the average distance in the MC(r, k) graphs. The multiplicative circulant graphs are vertex symmetric and have Hamiltonian cycles, one of them being 0, I. _. , n -1.  ...  Acknowledgements The author is grateful to two referees for careful reading which has greatly improved the clarity of the paper.  ...

### Random Walks, Bisections and Gossiping in Circulant Graphs

Bernard Mans, Igor Shparlinski
2013 Algorithmica
Using number theoretical tools, we first prove two main results for random directed k-regular circulant graphs with n vertices, when n is sufficiently large and k is fixed.  ...  First, for any fixed ε > 0, n = p prime and L ≥ p 1/k (log p) 1+1/k+ε , walks of length at most L terminate at every vertex with asymptotically the same probability.  ...  Acknowledgements The authors are very grateful to the referees for constructive and thorough comments. The research of B. M. by Australian Research Council Grant DP110104560, and that of I. E.  ...

### Self-stabilizing Population Protocols [chapter]

Dana Angluin, James Aspnes, Michael J. Fischer, Hong Jiang
2006 Lecture Notes in Computer Science
A protocol to construct a spanning tree in regular graphs using O(log D) memory is also given, where D is the diameter of the graph.  ...  Constant-space protocols are given for leader election in rings, local-addressing in degree-bounded graphs, and establishing consistent global direction in an undirected ring.  ...  The existence of a uniform constant-space leader election protocol on the class of all rings or on the class of regular communication graphs of degree d > 2 is still open for future research.  ...
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