1 Introduction
Analysis of regular graphs for their properties, including eigenspectra and automorphisms, is a fertile field for discovering and applications in algebraic graph theory Godsil2001 . Yet, there are many unsolved problems, e.g., the Conway’s 99graph problem conway and the 57regular Moore graph Hoffman1960 . For analysis of interconnection networks, regularity is essential for its direct and useful relationship to the complexity of network implementation, and as such, many regular graphs including the Peterson graph, hypercube graph, and their extensions Dasa ; Ohringa ; Ohring1993 ; Seo2011 ; Seo2017 ; Seo2008 are widely used to construct interconnection networks for parallel computers.
For regular graphs of order , Robinson and Wormald Robinson1983 ; Robinson1977 presented all counting results for , while pointing out that enumeration for unlabeled regular graphs with is an unsolved problem. Meringer Meringer1999 proposed a practical method to construct regular graphs without pairwise isomorphism checking but with a fast test for canonicity. Brankmann Brinkmann1996 ; Brinkmann2017 developed minibaum and snarkhunter for generating 3regular graphs. “The House of Graphs” Brinkmann2013 and the OEIS inc2019line databased the latest results for numbers of regular graphs. Kimberley A068934 contributed many results (A068934) to these databases by a package called GENREG developed by Meringer Meringer1999
. In addition to its challenges of pure mathematics, the graph counting problem is the root of topics in reliability, artificial intelligence, reasoning, statistical physics
Vadhan2001 , life sciences, chemistry Meringer2010 and even the search for the origins of life Meringer2017 .GENREG, efficient for smallscale clusters due to its feature of task partition, approaches a hard wall of speedup for finegrained partitioning on largescale clusters, caused mainly by load imbalance. For obtaining larger graphs, we extend GENREG for distributed clusters by using the message passing interface (MPI) mpi2014 . Using the parallel GENREG we developed, we have obtained the following results:

filtered all 3regular graphs up to order 32 with minimum ASPLs;

discovered thousands of 4regular graphs of order 32 with minimum ASPLs;

generated the exact counts of 4regular graphs of order 23 by using the three supercomputer clusters located in the US, China, and Ecuador.
Among our results, the first and second have been applied to the interconnection network research Deng2019 for benchmark relationship of the graph ASPLs to network performance latencies. The third expands from 22 to 23, for the first time, in the sequence A006820 a006820 of OEIS, which is the number of connected 4regular graphs of order . Kimberley a006820 used GENREG to enumerate the 4regular graphs for up to the order 22 in 2011 Larrion2016 . This record for remained unchallenged until our enumeration for , enabled by our parallel computing implementation to advance it a step.
2 The enumeration framework and results
2.1 The enumeration function
For enumerating the regular graphs, published packages such as minibaum Brinkmann1996 , snarkhunter Brinkmann2017 , and GENREG have their own strengths and weaknesses, GENREG is more general in covering the graph degrees than minibaum and snarkhunter that only support 3regular graphs.
In our parallel computing framework, we designate one node as the master, whose task involves adaptive scheduling and dispatching, and the rest as a team of workers. When our program starts, the workers send a message to the master to request a task, and the workers continue the requests until the list of tasks exhausts. As usual, when the master sends a task to a worker, this task is marked as selected and becomes unavailable. At last, when the task pool empties, the master signals all workers to exit.
Our dynamical scheduling strategy keeps cores in the cluster busy for useful tasks to allow us efficient search for graphs with specified parameters, e.g., diameters or eigenvalues, by inserting external serial programs. In addition to load balance, our parallel program reduces the communication cost to
, less requirement of bandwidth because the message itself is the message count. If we use a dedicated thread for task scheduling, the scalability and limit the maximum computer system can both shrink. Depending on the communication subsystem, the maximum scalable system our current approach can reach is approximately 3,000 cores, due to communication congestions, eventually. We may improve the scalability of our program by a multilevel scheduling; particularly, for the manycore systems.2.2 Search for a regular graph with minimal ASPL
In the interconnection networks of supercomputers and data centers, regularity is a very significant feature because it is related to the complexity of the network configuration. For the topologies of regular graphs applied to the interconnection networks, it is highly desirable to obtain graphs with minimal ASPLs because they help reduce communication latencies. Let be the distance between vertex and , the ASPL is calculated as follows,
(1) 
Cerf Cerf1975
calculated and proved lower bound of ASPL, hence the goal of optimal graph is to find graphs with minimum ASPLs. Usually and thus far, random or heuristic methods or intuitions were resorted to searching for graphs with such desired properties for large networks; Graph Golf
golf , a competition of searching for graphs with the minimal diameters and ASPLs, generated many graphs that are very commonly, but asymmetrically. However, we still hope to discover graphs with all desired properties including minimum diameters, ASPLs, symmetry, and robustness but with one disadvantage: smaller graphs. However, these graphs can be adopted in smaller clusters or multiple modules of these clusters or systemonchip.Using this framework, we decomposed the search into 200,000 subtasks and completed the exhaustive search of all of 3regular graphs of order 32 with diameter 4 and minimal ASPL. The program running on the SunwayBluelight supercomputer sunway with 80,000 cores for 72 hours and discovered 56 graphs with the minimal ASPLs after exhausting all 18,941,522,184,590 possible graphs predicted by Robinson et al. [10] who only enumerated them without finding the graphs with desired properties including minimum diameters and ASPLs. Deng et al. Deng2019 applied one (Fig. 1 (a)) of these graphs to construct a Beowulf BE2003 cluster, and their benchmark results show that the graphs with the minimal ASPL outpeform other classical topologies.
Fig. 2
shows the distribution of generated graph numbers for all cases, which look like Gaussian distribution with the long tail containing hundreds of millions of graphs and the frequency fluctuates by 4 orders of magnitude. Clearly, the tasks of enumerations vary greatly from graph to graph, and our dynamic task scheduling has eliminated substantial waiting time due to load imbalance.
2.3 Graph counting for (23,4)regular graphs
When we search for the minimumASPL graphs among all possible 4regular graphs of order 32, there is no enumeration result for this scale when . We can still use our software to search for regular graphs with minimum ASPL, as shown in Fig. 1 (b), without confirmation of exhaustiveness. Therefore, we verified the results in Tab. 1 after decomposing the problem into 50,000 subtasks and setting the splitlevel as 12. We manage to increase the of the sequence A006820 from 22 to 23 and confirmed the number of 4regular graphs of order 23 as 429,668,180,677,439 by using our parallel GENREG with the same parameters.
Order  Quartics 

5  1 
6  1 
7  2 
8  6 
9  16 
10  59 
11  265 
12  1,544 
13  10,778 
14  88,168 
15  805,491 
16  8,037,418 
17  86,221,634 
18  985,870,522 
19  11,946,487,647 
20  152,808,063,181 
21  2,056,692,014,474 
22  28,566,273,166,527 
23  429,668,180,677,439 
Our work to obtain the new enumeration for
is estimated to cost nearly 100 coreyears. We ganged three supercomputers, the SeaWulf at Stony Brook University, while the Tianhe1 with Intel Xeon X5670 processors and the IBM Quinde with Power8 processors can process 113,000 and 56,469 graphs per second per core, respectively. As shown in Tab.
2, the Seawulf with Intel Xeon Gold 6148 processors has the highest efficiency for searching 178,000 graphs per second per core, while the Tianhe1 with Intel Xeon X5670 processors and the IBM Quinde with Power8 processors contributed the rest.In fact, the result of graph counting for (23,4)regular graphs was obtained by the strategic and opportunistic use of the fragmented and shared computing resources. In addition to the policy of fair and efficient share for most supercomputers, the Tianhe1 would terminate tasks that last long for many cores because of maintenance, which prevents us from running long tasks. The external scheduling system we developed helps overcome the limitations of computing resources while facilitating optimal utilization of the occasionally available cores.
Cluster 





Seawulf  66.12  178  
Tianhe1  19.53  113  
IBM Quinde  13.25  56 
3 Conclusions
Our parallel method adapting the GENREG enables us to complete the research and enumeration on systems of 3000 processor cores. For the first time, using this new approach, we discovered several optimal graphs of order 32 and found the enumeration count for 4regular graphs of order 23, gaining confidence in the graph theory community that highperformance computing can help solve otherwise intractable problems.
Acknowledgements
Z. Xu is supported by the special project high performance computing of National Key Research and Development Program 2016YFB0200604. We thank the National Supercomputing Centers in Jinan and Changsha in China, for computing resources and, M. Meringer of German Aerospace Center, for technical support and of GENREG and beneficial suggestions of this manuscript via emails.
References
 (1) C. Godsil, G. Royle, Algebraic Graph Theory, Springer New York, 2001. doi:10.1007/9781461301639.
 (2) J. Conway, Five $1,000 problems (update 2017), OnLine Encyclopedia of Integer Sequences (2019).
 (3) A. J. Hoffman, R. R. Singleton, On moore graphs with diameters 2 and 3, IBM Journal of Research and Development 4 (5) (1960) 497–504. doi:10.1147/rd.45.0497.
 (4) S. Das, A. Banerjee, Hyper petersen network: yet another hypercubelike topology, in: [Proceedings 1992] The Fourth Symposium on the Frontiers of Massively Parallel Computation, IEEE Comput. Soc. Press, 1992. doi:10.1109/fmpc.1992.234949.
 (5) S. Ohring, S. Das, Folded petersen cube networks: new competitors for the hypercubes, in: Proceedings of 1993 5th IEEE Symposium on Parallel and Distributed Processing, IEEE Comput. Soc. Press, 1993. doi:10.1109/spdp.1993.395482.
 (6) S. Ohring, S. K. Das, The folded petersen network : A new communicationefficient multiprocessor topology, in: 1993 International Conference on Parallel Processing  ICPP 93 Vol1, IEEE, 1993. doi:10.1109/icpp.1993.175.
 (7) J.H. Seo, Threedimensional petersentorus network: a fixeddegree network for massively parallel computers, The Journal of Supercomputing 64 (3) (2011) 987–1007. doi:10.1007/s112270110716z.
 (8) J.H. Seo, J.S. Kim, H. J. Chang, H.O. Lee, The hierarchical petersen network: a new interconnection network with fixed degree, The Journal of Supercomputing 74 (4) (2017) 1636–1654. doi:10.1007/s1122701721864.
 (9) J.H. Seo, H. Lee, M. suk Jang, Petersentorus networks for multicomputer systems, in: 2008 Fourth International Conference on Networked Computing and Advanced Information Management, IEEE, 2008. doi:10.1109/ncm.2008.47.
 (10) R. W. Robinson, N. C. Wormald, Numbers of cubic graphs, Journal of Graph Theory 7 (4) (1983) 463–467. doi:10.1002/jgt.3190070412.
 (11) R. W. Robinson, Counting cubic graphs, Journal of Graph Theory 1 (3) (1977) 285–286. doi:10.1002/jgt.3190010310.
 (12) M. Meringer, Fast generation of regular graphs and construction of cages, Journal of Graph Theory 30 (2) (1999) 137–146. doi:10.1002/(sici)10970118(199902)30:2<137::aidjgt7>3.0.co;2g.
 (13) G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory 23 (2) (1996) 139–149. doi:10.1002/(sici)10970118(199610)23:2<139::aidjgt5>3.0.co;2u.
 (14) G. Brinkmann, J. Goedgebeur, Generation of cubic graphs and snarks with large girth, Journal of Graph Theory 86 (2) (2017) 255–272. doi:10.1002/jgt.22125.
 (15) G. Brinkmann, K. Coolsaet, J. Goedgebeur, H. Mélot, House of graphs: A database of interesting graphs, Discrete Applied Mathematics 161 (12) (2013) 311–314. doi:10.1016/j.dam.2012.07.018.

(16)
OEIS Foundation Inc, The online encyclopaedia of
integer sequences (2019).
URL http://oeis.org 
(17)
OEIS Foundation Inc,
A068934 in the
online encyclopaedia of integer sequences (2019).
URL http://oeis.org/wiki/User:Jason_Kimberley/A068934  (18) S. P. Vadhan, The complexity of counting in sparse, regular, and planar graphs, SIAM Journal on Computing 31 (2) (2001) 398–427. doi:10.1137/s0097539797321602.
 (19) M. Meringer, Structure enumeration and sampling, in: Handbook of Chemoinformatics Algorithms, Chapman and Hall/CRC, 2010, pp. 233–267. doi:10.1201/9781420082999c8.
 (20) M. Meringer, H. J. Cleaves, Exploring astrobiology using in silico molecular structure generation, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375 (2109) (2017) 20160344. doi:10.1098/rsta.2016.0344.
 (21) W. Gropp, E. Lusk, A. Skjellum (Eds.), Using MPI: Portable Parallel Programming with the Message Passing Interface, MIT Press Ltd, 2014.
 (22) Y. Deng, M. Guo, A. F. Ramos, X. Huang, Z. Xu, W. Liu, Optimal lowlatency network topologies for cluster performance enhancement, arxiv (2019). arXiv:http://arxiv.org/abs/1904.00513v1.

(23)
OEIS Foundation Inc, A006820 in the online
encyclopaedia of integer sequences (2019).
URL http://oeis.org/A006820 
(24)
F. Larrión, M. Pizaña, R. VillarroelFlores,
On selfclique shoal
graphs, Discrete Appl. Math. 205 (C) (2016) 86–100.
doi:10.1016/j.dam.2016.01.013.
URL https://doi.org/10.1016/j.dam.2016.01.013  (25) V. G. Cerf, D. D. Cowan, R. C. Mullin, R. G. Stanton, A partial census of trivalent generalized moore networks, in: Combinatorial Mathematics III, Springer Berlin Heidelberg, 1975, pp. 1–27. doi:10.1007/bfb0069540.
 (26) M. Koibuchi, I. Fujiwara, S. Fujita, K. Nakano, T. I. T. Uno, K. Kawarabayashi, Graph Golf: The Order/degree Problem Competition, http://research.nii.ac.jp/graphgolf/.

(27)
The Sunway Blue Light in Top 500
List (June 2018).
URL https://www.top500.org/system/177447  (28) W. Gropp, E. L. Lusk, T. Sterling (Eds.), Beowulf Cluster Computing with Linux (Scientific and Engineering Computation), The MIT Press, 2003.
Comments
There are no comments yet.