1 Introduction
In graph theory, a regular graph is a connected graph that each vertex has the same number of neighbors; i.e., each node has the same amount of edges. The research on regular graphs attracted attention widely for a long time ago. Some properties (e.g., eigenvalue, group of automorphisms) of regular graphs have been discussed deep in algebraic graph theory
Godsil2001 . Until now, there also exist some interesting unsolved problems in this area, e.g., Conway’s 99graph problemconway , 57regular Moore graphHoffman1960 .In the field of interconnection networks, regularity is an essential feature because it is related to the complexity of network implementation. Therefore, some classical regular graphs (e.g., Peterson graph, cubic graph) and their extensionsDasa ; Ohringa ; Ohring1993 ; Seo2011 ; Seo2017 ; Seo2008 are used to construct interconnection network. Besides, the regular graph is also related to coding theoryRybin2018 ; Rosenthala
and machine learning
VegaOliveros2014 , etc.Suppose that there are vertices, McKay and WormaldMcKay1991 proved that the number of regular graph for order with is asymptotically
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Furthermore, they reduced the degree to for constant
and proposed new estimated results
McKay1990 . Then LiebenauLiebenau2017 filled the gap of two degrees. However, if we want to get the exact number of regular graphs with order , we should construct all nonisomorphic graphs and count them. MeringerMeringer1999 proposed a practical method to construct regular graphs without isomorphism checking that combined with a fast test for canonicity. Besides, BrankmannBrinkmann1996 ; Brinkmann2017 developed minibaum and snarkhunter for generating cubic graphs. Also, the website of ’House of Graphs’Brinkmann2013 and the OEIS database recorded the latest results for counting of the regular graph, Jason Kimberley contributed many results (A068934) by using GENREG as well. However, the GENREG support singlecore because the core algorithm is recursive and it is hard to accelerate after parallelization, which means it is not matching for the cluster in high efficiency though dividing the task for many parts. In the branch of ’cubic graph’ in the database of ’House of Graphs,’ they mentioned that Jason Kimberley could run GENREG up to 250 cores in 2009, but it is not enough to adapt to the incredible increase of core number for modern supercomputers.Hence, in this note, we introduced the method to extend the GENREG for the distributed cluster that can support thousands of cores with the message passing interface(MPI), which is popular in highperformance computing, and we named it as DistriGENREG. Using the modified version of the regular graph generation, we achieved some results as follows

find all of 55 graphs with the minimum diameter in the 3regular graphs for order 32;

got thousands of 4regular graphs for order 32 with the minimum diameter 3;

exhausted all of the 4regular graphs for order 23 and got the exact counting number that is 429668180677439.
In the list, the first and second results have been applied to the interconnection network researchDeng2019 because the graph diameter will affect the network performance. The third result expands from 22 to 23 in the sequence A006820 of OEIS, which is the number of connected regular simple graphs of degree 4 (or quartic graphs) with
nodes; actually, the graph counting problem is closely related to reliability, artificial intelligence, reasoning, and statistical physics
Vadhan2001 .2 The framework and results
2.1 Function
For enumerating or counting the regular graphs, we compared minibaum, snarkhunter, and GENREG, and found that GENREG can support more range of degrees, but minibaum and snarkhunter only support the 3regular graph. Here, we did not modify the core algorithm and codes but add a layer of the wrap to do schedule for parallel computing.
In this framework, we divide the computing nodes into one master thread and some workload threads. The function of the master thread is to maintain the task schedule and arrange the job. When the program starts, the workload thread sends a message to the master thread to request a task, then request again if one task finished. Furthermore, when master thread sends the task to workload thread, this task would be marked as ’pass’ and not be sent again. At last, the master thread sends an individual packet to inform the workload thread to exit when there is no task to arrange.
After adapting the dynamical schedule strategy, that can make all of the cores in the cluster keep working. Meanwhile, we can search for some graphs with specified parameters (e.g., diameter, eigenvalue) with inserting external codes. Besides, the communication cost of this strategy is , which costs less bandwidth because the message is the part number. If we use one thread to schedule tasks, the maximum parallel scale is cores due to communication congestion and the precise value is depending on the hardware configuration, but it is adequate for most clusters. Furthermore, we can improve the parallel scale by using multilevel scheduling, and this is effective, especially for the manycore system.
2.2 Search of the (32,3)regular graph with minimal diameter
In the interconnection network of supercomputer and data center, regularity is a very significant feature because it is related to the complexity of the network configuration. For the regular networks that applied to the interconnection network, the minimal diameter and average shortest path length(ASPL) are desired parameters because the low diameter or ASPL can reduce communication latency. We usually use the random or heuristic method to search for the graphs with desired properties for large scale network and Gold Graph is a competition to search for the optimal graph with the minimal diameter and ASPL, but they are asymmetric in most conditions. However, we still hope to get all of the graphs with the minimal diameter, after filtering with specified properties (e.g., symmetry, robustness) can be applied to the small scale cluster or systemonchip.
By using this framework, we split all tasks into 200,000 parts and got all of the 3regular graphs for order 32 with diameter 4. The program launched on ShenweiBluelight supercomputer with cores for hours and picked up 55 graphs with diameter 4 after exhausted 18941522184590 graphs that agree with RobinsonRobinson1983 , then DengDeng2019 applied one(Fig.2(a)) of these graphs to the beowulfBE2003 cluster named as ’Taishan’ and the benchmark results show that the graph with the minimal diameter can enhance the performance than other classical topologies. Fig.2(b) shows the distribution of generated graph numbers for all parts, which look like tailed Gauss distribution and most parts contain graphs and the number fluctuated in an extensive range. Therefore if we do not introduce dynamical scheduling but using static task arrangement, we should spend more time to get these results.
2.3 Graph counting for (23,4)regular graphs
When we plan to find all of the 4regular graphs with the minimum diameter for order 32, it is unlucky there is no counting result for this scale when , but we still can use this software to search for the regular graphs with small diameter as shown in Fig2(a) though we can’t see the endpoint the road. Therefore, we verified the results in Table.1 after splitting the problem into 50,000 parts and setting the splitlevel as 12, then promoted of the sequence A006820 from 22 to 23 and confirmed the number of 4regular graphs for order 23 is 429,668,180,677,439 by using DistriGENREG with same parameters. RobinsonRobinson1983 presented all of the counting numbers of 3regular graphs up to order 40 in 1983, but he also pointed out that the enumeration of regular is complicated and there did not exist an efficient method. Then Markus Meringer and Jason Kimber used the GENREG to generate and count up the quartic graphs up to order 22 in 2011Larrion2016 , and this problem remained stagnant for many years. With the development of highperformance computing, that provides the chance that we can refocus and promote the problem by using more powerful supercomputers.
Order  Quartics  Order  Quartics 

5  1  14  88,168 
6  1  15  805,491 
7  2  16  8,037,418 
8  6  17  86,221,634 
9  16  18  985,870,522 
10  59  19  11,946,487,647 
11  265  20  152,808,063,181 
12  1,544  21  2,056,692,014,474 
13  10,778  22  28,566,273,166,527 
23  429,668,180,677,439 
This work cost almost 99 coreyear, and SeaWulf, Tianhe1A, Yachay joined in this work. Fig.3 presents the computing time for each cluster that pure CPU nodes (Intel Xeon Gold 6148 2) in Seawulf contribute most results, Tianhe1A and Yachay (IBM Power8 rev 2.0) also work well for this program, but we didn’t use GPU in these two machines. Also, the task parts for the three clusters overlapped in a small range to prevent the impact from different systems.
3 Conclusions
In this paper, we presented the method to transport the GENREG to cluster and support the thousands of CPU cores. Then, we used it to got some optimal graphs for order 32 and calculated the exact number for 4regular graphs for order 23. Above all, we can see that this framework is useful to solve intractable problems for the regular graph. Finally, we also decide to open our source code in GitHub(https://github.com/xuzhipengnt/distriGENREG) to attract more researchers to join in the area of regular graph counting.
4 Acknowledgement
In this work, the first author was supported by the special project High performance computing of National Key Research and Development Program 2016YFB0200604. We thank the Computing Center of Stony Brook University(USA), National Supercomputing Center in Jinan & Changsha(China) and National Supercomputing Center(Ecuador) for support of computing resources. Thanks to Dr. Markus Meringer from German Aerospace Center for technical support for GENREG by email.
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