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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. ...arXiv:1506.04962v3 fatcat:4q2b6tj3obd5dlodvdxd77htsy
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. ...doi:10.26493/1855-3974.969.659 fatcat:v6x4yqlh4bhzngd4bebvpxyyga
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. ...arXiv:1404.3948v3 fatcat:zrhv5edsqfa43c7f4bddp32ewy
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. ...doi:10.37236/4279 fatcat:7xutpzr7t5aphhapmeqm6vxhyq
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. ...
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 degree ∆ and 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 degree ∆ and diameter D. ...doi:10.5614/ejgta.2014.2.2.9 fatcat:e57fyrwwojao3aayms3r2yz73y
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. ...arXiv:1506.02844v1 fatcat:dfnppjx5mzexri7yczfnz7mbqe
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. ...
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. ...doi:10.1137/1.9781611975031.170 dblp:conf/soda/KosowskiU18 fatcat:vkj5kb2cojbvrcn22vlc3ibdoe
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. ...arXiv:1612.09145v3 fatcat:wwiivx74dngu7lts56ufhqxlwe
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. ...doi:10.12783/dtcse/cmsam2016/3594 fatcat:it2hj4vnqjcuvhghtweaxgtyrm
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. ...doi:10.1016/j.jctb.2018.07.002 fatcat:fo3sd3ppungehotggzboiiglwa
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. ...doi:10.1016/s0166-218x(96)00138-2 fatcat:g7dt3dwhanhhtjb3v7l6mu3xhq
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. ...doi:10.1007/s00453-013-9810-3 fatcat:34x4xvpd3fhqxew5ckujy57doe
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. ...doi:10.1007/11795490_10 fatcat:ojvtg73gzvcmfggr2wnqwc6mhm
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