Construction of Some Circular Regular Graph Designs in Blocks of Size Four Using Cyclic Shifts

Rashid Ahmed; Farrukh Shehzad; Muhammad Jamil; H. M. Kashif Rasheed

doi:10.2991/jsta.d.200423.001

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Volume 19, Issue 2, June 2020, Pages 314 - 324

Construction of Some Circular Regular Graph Designs in Blocks of Size Four Using Cyclic Shifts

Authors

Rashid Ahmed¹^{, *}, Farrukh Shehzad¹, Muhammad Jamil², H. M. Kashif Rasheed¹

¹Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan

²Allama Iqbal Open University Islamabad, Islamabad, Pakistan

^*Corresponding author. Email: rashid701@hotmail.com

Corresponding Author

Rashid Ahmed

Received 3 January 2018, Accepted 16 March 2020, Available Online 6 July 2020.

DOI: 10.2991/jsta.d.200423.001 How to use a DOI?
Keywords: Balanced incomplete block designs; Block designs; Partially balanced incomplete block designs; Regular graph designs
Abstract: Circular regular graph designs play an important role in the design of experiments where most of the balanced incomplete block designs require a large number of blocks. In this article, circular regular graph designs are constructed in blocks of size four through cyclic shifts. Without studying the complete design, some standard properties of the designs can be observed only through the sets of shifts. Therefore, method of cyclic shifts has an edge over existing methods.
Copyright: © 2020 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Block designs are used in experimental planning with the purpose of maximizing the information extracted from a given number of experiments. If homogeneous blocks of size k are available to accommodate all the k treatments, a randomized complete block design is preferred. Incomplete block designs are used in situations where all the treatment combinations could not be run in each block. The most popular incomplete block designs are balanced incomplete block designs (BIBDs) introduced by Yates [1]. BIBDs compare all treatments pairs with equal precision. As the class of BIBDs do not fit for many experimental situations because often these designs require a large number of replications, to overcome this Bose and Nair [2] introduced a class of binary, equireplicate and proper designs called partially balanced incomplete block designs (PBIBDs). Bose [3] established the relation between PBIBDs and strongly regular graphs. Bose and Shimamoto [4] are first to introduce the concept of association scheme in PBIBDs. Bose [5] used the graph theoretic method for the study of association schemes of PBIBDs and also shown that strongly regular graph emerges from PBIBD with two associate class.

A PBIBD is obtained by identifying the v treatments with the v objects of an association scheme arranging into b blocks satisfying the following conditions:

Each block contains k treatments.
Each treatment occurs in r blocks.
If two treatments are ith associates, they occur together in λi blocks.
Each treatment has exactly niith associates.
Given any two treatments which are ith associates, the number of treatments common to the jth associates of the first, and the kth associates of the second is and is independent of the pair of treatments.

An associate class is a set of treatment pairs where each pair from the set occur together the same number of times, λi. Regular graph design (RGD) is an important class of PBIB designs with two association scheme. A RGD (v, k, r) is a collection of blocks of size k on a v-set (with no restriction on repeated blocks) such that every element occurs in r blocks and any pair of objects occur together in either λ₁ or λ₂ blocks, where λ₁ is some constant and λ2=λ1+1 RGDs were introduced by John and Mitchell [6]. Kreher et al. [7] discussed the existence of resolvable RGDs with block size 4, 8, 12 and 16 points. A design is resolvable if its set of blocks can be partitioned into r parallel classes or resolutions such that each element occurs in exactly one block in each class. Barandag et al. [8] considered the association scheme which is related to the flag algebra of a BIBD, with λ=1. By finding a suitable equivalence of this scheme, they constructed a 2-class association scheme. Moreover, each 2-class association scheme is equivalent to a strongly regular graph. Cakiroglu [9] constructed optimal RGDs by pjki adding the blocks of a BIBD repeatedly to the original design and presented the best RGDs for v≤20, k≤10 and replication r≤10.

Nair and Rao [10] have developed a set of sufficient combinatorial conditions which lead to construction of confounded designs. A catalogue of different PBIB on two associate class designs can be found in Clatworthy [11]. Cheng and Wu [12] constructed nearly BIBDs. Wallis [13] discussed measures of optimality of RGDs from a combinatorial viewpoint. Various properties of these designs were also discussed. Kumar [14] has given the construction of PBIBDs through unreduced BIBDs. Waliker et al. [15] have established the relation between minimum dominating sets of a graph with the blocks of PBIBDs. Using method of cyclic shifts, Yasmin et al. [16] constructed some classes of BIBDs. In this study, RGDs are constructed in blocks of size four through method of cyclic shifts.

2. METHOD OF CYCLIC SHIFTS

Method of cyclic shifts introduced by Iqbal [17] is simplified here only for BIBDs, PBIBDs and RGDs. In this construction, v treatments are labeled as 0,1,2,…,v−1. For further detail, see Yasmin et al. [16].

Let Sj=qj1,qj2,…,qj (k−1) be set(s) of shifts where 1≤qji≤v−1. A design is BIBD if each element of Sj∗ contains all elements 1,2,…,v−1, equal number of times, say, λ. Where Sj∗=qj1,qj2,…,qj (k−1), qj1+qj2,qj2+qj3,…, (qj(k−2)+qj(k−1)),(qj1+qj2+qj3), (qj2+qj3+qj4),…, qj (k−3)+qj (k−2)+qj (k−1),…, qj1+qj2+…+qj (k−1), v−qj1,v−qj2,…,v−qj k−1, v−qj1+qj2,v−qj2+qj3,…, v−qj (k−2)+qj (k−1), v−qj1+qj2+qj3, v−qj2+qj3+qj4,…, v−qj (k−3)+qj (k−2)+qj (k−1),…, v−(qj1+qj2+…+qj (k−1)). If λ has two values as λ₁ and λ2=λ1+1 then it is RGD (PBIBD with two associate class).

Example 2.1

RGD is constructed from the set of shifts [2, 1, 4] for v=9 and k=4 with λ1=1 and λ2=2.

Here S=2,1,4, v=9 and k=4 then S∗=2,1,4,3,5,7,7,8,5,6,4,2 contains each of 1, 2, …, 8 either once or twice, according to the method of cyclic shifts, it is a RGD. Now we explain the procedure to complete the design from the given set of shifts [2, 1, 4].

Consider 0, 1, …, and 8 as the elements of first unit for all blocks. To get the elements of second units for all blocks, add 2 (mod 9) to the each element of first unit for all blocks. To get the elements of third units for all blocks, add 1 (mod 9) to the each element of second unit for all blocks. Similarly add 4 (mod 9) to get the elements of fourth units for all blocks.

B₁	B₂	B₃	B₄	B₅	B₆	B₇	B₈	B₉
0	1	2	3	4	5	6	7	8
2	3	4	5	6	7	8	0	1
3	4	5	6	7	8	0	1	2
7	8	0	1	2	3	4	5	6

3. CONSTRUCTION OF RGDS IN BLOCKS OF SIZE FOUR

In this section, RGDs are constructed in blocks of size four through following i sets of cyclic shifts.

Sj=sj 1,sj 2,sj 3; j=1,2,…,i.

Such that

1≤sj 1,sj 2,sj 3≤v−1,
sj 1+sj 2+sj 3 mod v≠0,
For λ1=0 and λ2=1, each of 1,2,…,v−1 appears once or no time in S*.
For λ1=1 and λ2=2, each of 1,2,…,v−1 appears once or twice in S*.
S∗={sj 1,sj 2,sj 3,sj 1+sj 2,sj 2+sj 3,sj 1+sj 2+sj 3,v−sj 1,v−sj 2,v−sj 3, v−(sj 1+sj 2),v−(sj 2+sj 3),v−(sj 1+sj 2+sj 3) mod v}

In these designs, any pair of treatments which are first associates occur in exactly λ₁ blocks and second associates occur together in exactly λ₂ blocks, together. RGDs have eight parameters (v, b, r, k, λ₁, λ₂, n₁, n₂) of kind I and six parameters (pjki,i,j,k=1,2) of kind II. The parameters of kind II may be arranged in the form of two symmetric matrices (P-matrices).

P1=p111p121p211p221 and P2=p112p122p212p222

RGDs constructed in these series are cyclic, therefore, general expression of P matrices can be written as

P1= α n1−α−1n1−α−1n2−n1+α+1 and P2= β n1−βn1−βn2−n1+β−1

where p111=α is the number of treatments common to first associates of two treatments. Where these two treatments are first associates of each other. Similarly, p112=β is the number of treatments common to first associates of two treatments. Which these two treatments are second associates of each other.

Series 3.1: RGDs can be constructed for v=6w+3, k=4, b=vw, r=4w with n1=4, n2=v−5, λ1=1 and λ2=2 through w sets of shifts. P-matrices for these designs are

P1= α 3−α3−α10−v+α and P2= β 4−β4−β8−v+β

Example 3.1

Set of shifts [2, 1, 4] provides RGD for v=9, k=4 with n1=n2=4, λ1=1 and λ2=2.

P-matrices for this design are P1=0331 and P2=3112

For the convenience of readers, first and second associates of all treatments for this design are arranged in the following table:

Treatment No.	First Associates	Second Associates
0	1, 3, 6, 8	2, 4, 5, 7
1	0, 2, 4, 7	3, 5, 6, 8
2	1, 3, 5, 8	0, 4, 6, 7
3	0, 2, 4, 6	1, 5, 7, 8
4	1, 3, 5, 7	0, 2, 6, 8
5	2, 4, 6, 8	0, 1, 3, 7
6	0, 3, 5, 7	1, 2, 4, 8
7	1, 4, 6, 8	0, 2, 3, 5
8	0, 2, 5, 7	1, 3, 4, 6

Here, n1=n2=4 and the parameters of kind II are

p111=α=0, p112=β=3, p121=p211=n1−α−1=4−0−1=3,p221=n2−n1+α+1=4−4+0+1=1,p122=p212=n1−β=4−3=1,p222=n2−n1+β−1=4−4+3−1=2P1=0331,P2=3112

Consider two treatments that are first associates of each other, treatment 0 and 1 are first associates; there is no treatment common in first associates of treatment 0 and 1, hence p111=α=0. Similarly, there are three treatments 3, 6, 8 (or 2, 4, 7) that are common in first associates of treatment 0 and second associates of treatment 1 (or common in second associates of treatment 0 and first associates of treatment 1), then p121=p211=3. And there is only one treatment that is common in second associates of treatment 0 and 1 which is 5, therefore, p221=1. Similarly, P₂-matrix can be constructed.

Catalogue of RGDs constructed under Series 3.1 is presented in Table A.1 of Appendix A.

Series 3.2: RGDs can be constructed for v=12w+5, k=4,b=vw,r=4w with n1=4, n2=v−5, λ1=1 and λ2=0 through w sets of shifts. P-matrices for these designs are

P1= α 3−α3−α10−v+α and P2= β 4−β4−β8−v+β

Example 3.2

Set of shifts [1, 2, 4] provides RGD for v=7,k=4,b=17,r=4 with n1=4,n2=12,λ1=1 and λ2=0.

P-matrices for this design are P1=0339 and P2=1338

Catalogue of RGDs constructed under Series 3.2 is presented in Table A.2 of Appendix A.

Series 3.3: RGDs can be constructed for v=12w−1, k=4,b=vw,r=4w with n1=v−3, n2=2,λ1=1 and λ2=2 through w sets of shifts. P-matrices for these designs are

P1= αv−α−4v−α−4α−v+6 and P2= βv−β−3v−β−3β−v+4

Example 3.3

Set of shifts [1, 2, 4] provides RGD for v=11,k=4,b=11,r=4 with n1=8,n2=2,λ1=1 and λ2=2.

P-matrices for this design are P1=5220 and P2=7110

Following are RGDs for v = 11 and 23

Parameters							Sets of Shifts
v	b	r	n₁	n₂	α	β
11	11	4	8	2	5	7	[1, 2, 4]
23	46	8	20	2	17	19	[1, 2, 7] + [5, 6, 8]

Series 3.4: RGDs can be constructed for v=12w+5, k=4, b=vw+1, r=4w+1 with n1=v−9, n2=8, λ1=1 and λ2=2 through w+1 sets of shifts. P-matrices for these designs are

P1= αv−α−10v−α−10α−v+18 and P2= βv−β−9v−β−9β−v+16

Example 3.4

Sets of shifts [1, 2, 4] and [5, 8, 3] provide RGD for v=17,k=4,b=34,r=8 with n1=8,n2=8,λ1=1 and λ2=2.

P-matrices for this design are P1=4335 and P2=3552

Catalogue of RGDs constructed under Series 3.4 is presented in Table A.3 of Appendix A.

Series 3.5: RGDs can be constructed for v=12w−4,k=4,b=vw,r=4w with n1=v−6,n2=5,λ1=1 and λ2=2 through w sets of shifts. P-matrices for these designs are

P1= αv−α−7v−α−7α−v+12 and P2= βv−β−6v−β−6β−v+10

Example 3.5

Set of shifts [2, 1, 4] provides RGD for v=8,k=4,b=8,r=4 with n1=2,n2=5,λ1=1 and λ2=2.

P-matrices for this design are P1=0114 and P2=0222

Catalogue of RGDs constructed under Series 3.5 is presented in Table A.4 of Appendix A.

Series 3.6: RGDs can be constructed for v=12w+2,k=4,b=vw+1, r=4w+1,n1=v−12 with n2=11,λ1=1 and λ2=2 through w+1 sets of shifts. P-matrices for these designs are

P1= αv−α−7v−α−7α−v+12 and P2= βv−β−6v−β−6β−v+10

Example 3.6

Sets of shifts [1, 2, 5] and [4, 1, 3] provide RGD for v=14,k=4,b=28,r=8 with n1=2,n2=11,λ1=1 and λ2=2.

P-matrices for this design are P1=01110 and P2=0228

Catalogue of RGDs constructed under Series 3.6 is presented in Table A.5 of Appendix A.

Series 3.7: RGDs can be constructed for v=12w−2,k=4,b=vw,r=4w with n1=v−4,n2=3,λ1=1 and λ2=2 through w sets of shifts. P-matrices for these designs are

P1= αv−α−5v−α−5α−v+8 and P2= βv−β−4v−β−4β−v+6

Example 3.7

Set of shifts [2, 1, 4] provides RGD for v=10,k=4,b=10,r=4 with n1=6,n2=3,λ1=1 and λ2=2.

P-matrices for this design are P1=2330 and P2=4220

Following are RGDs for v = 10 and 22:

Parameters							Sets of Shifts
v	b	r	n₁	n₂	α	β
10	10	4	6	3	2	4	[2, 1, 4]
22	44	8	18	3	14	16	[4, 1, 7] and [2, 6, 3]

Series 3.8: RGDs can be constructed for v=12w+4,k=4,b=vw+1,r=4w+1 with n1=v−10,n2=9,λ1=1 and λ2=2 through w+1 sets of shifts. P-matrices for these designs are

P1= αv−α−11v−α−11α−v+20 and P2= βv−β−10v−β−10β−v+18

Example 3.8

Sets of shifts [1, 2, 4] and [5, 8, 9] provide RGD for v=16,k=4,b=32,r=8, with n1=6,n2=9,λ1=1 and λ2=2.

P-matrices for this design are P1=2336 and P2=2444

Catalogue of RGDs constructed under Series 3.8 is presented in Table A.6 of Appendix A.

Series 3.9: RGDs can be constructed for v=12w+6,k=4,b=vw+1,r=4w+1 with n1=v−8,n2=7,λ1=1 and λ2=2 through w+1 sets of shifts. P-matrices for these designs are

P1= αv−α−9v−α−9α−v+16 and P2= βv−β−8v−β−8β−v+14

Example 3.9

Sets of shifts [1, 2, 4] and [5, 8, 9] provide RGD for v=18,k=4,b=36,r=8 with n1=10,n2=7,λ1=1 and λ2=2.

P-matrices for this design are P1=3661 and P2=6442

Catalogue of RGDs constructed under Series 3.9 is presented in Table A.7 of Appendix A

Following are some more RGDs for k=4.

Design 1. Set of shifts [1, 2, 4] provides RGD for v=12,k=4,b=12,r=4 with n1=10,n2=1,λ1=1 and λ2=2.

P-matrices for this design are P1=8110 and P2=10000

Design 2. Set of shifts [2, 8, 1] provides RGD for v=14,k=4,b=14,r=4 with n1=1,n2=12,λ1=1 and λ2=0.

P-matrices for this design are P1=00012 and P1=01010

Design 3. Set of shifts [2, 3, 4] provides RGD for v=15,k=4,b=15,r=4 with n1=2,n2=12,λ1=1 and λ2=0.

P-matrices for this design are P1=01111 and P1=01110

Design 4. Sets of shifts [3, 4, 5] and [21, 19, 25] provide RGD for v=27,k=4,b=54,r=4 with n1=24,n2=2,λ1=1 and λ2=0.

P-matrices for this design are P1=23110 and P1=22110

In Appendix B, designs constructed in this article are compared with the RGDs already available in literature.

4. CONCLUSION

Because of the importance of RGDs, it is much needed to have a comprehensive list/catalogues of this class of designs. Therefore, RGDs have been constructed in this article for blocks of size four through method of cyclic shifts. Nine series have been proposed along with some individual designs. Proposed designs have also been compared with the designs constructed by Bose et al. [18], John et al. [19] and Clatworthy [11]. Our proposed designs are new and have the efficiency greater than or equal to that of existing designs.

CONFLICT OF INTEREST

Certified that there is no conflict of interest.

AUTHORS' CONTRIBUTIONS

Certified that all the four authors contributed almost equally. Rashid Ahmed and Muhammad Jamil constructed RGDs for this article. Farrukh Shehzad and H. M. Kashif Rasheed calculated efficiencies for the comparison of these designs with those of existing designs.

Funding Statement

Certified that there is no funding agency for this research.

ACKNOWLEDGMENTS

Authors are thankful to the reviewers for their valuable suggestions and corrections.

APPENDIX A

v	r	n₁	n₂	β	Sets of Shifts
9	4	4	4	3	[2, 1, 4]
15	8	4	10	1	[1, 2, 4] + [2, 3, 4]
21	12	4	16	0	[1, 5, 3] + [1, 10, 4] + [2, 3, 4]
27	16	4	22	0	[1, 5, 6] + [2, 8, 9] + [4, 3, 13] + [1, 3, 9]
33	20	4	28	0	[2, 8, 3] + [1, 5, 12] + [4, 7, 9] + [4, 14, 5] + [1, 6, 2]
39	20	4	34	0	[2, 8, 3] + [1, 5, 12] + [4, 7, 9] + [4, 14, 5] + [3, 12, 2] + [8, 1, 6]
45	28	4	40	0	[1, 9, 5] + [2, 1, 19] + [2, 10, 15] + [3, 8, 16] + [4, 13, 6] + [5, 7, 9] + [6, 7, 4]
51	32	4	46	0	[1, 10, 12] + [3, 4, 14] + [5, 4, 16] + [6, 2, 13] + [12, 5, 19] + [1, 8, 2] + [13, 7, 17] + [19, 3, 23]

Table A.1

Catalog of regular graph designs (RGDs) under Series 3.1 with λ₁ = 1 and λ₂ = 2.

v	r	n₁	n₂	β	Sets of Shifts
17	4	4	12	1	[1, 2, 4]
29	8	4	24	1	[1, 2, 8] + [4, 5, 7]
41	12	4	36	0	[1, 5, 4] + [2, 11, 7] + [14, 3, 16]
53	16	4	48	0	[1, 9, 12] + [2, 15, 13] + [3, 4, 20] + [5, 6, 8]
65	20	4	60	1	[1, 9, 12] + [2, 15, 13] + [3, 4, 20] + [6, 5, 14] + [18, 16, 23]
77	24	4	72	0	[2, 10, 25] + [3, 11, 22] + [4, 5, 23] + [8, 21, 17] + [15, 1, 18] + [20, 6, 24]
89	28	4	84	0	[1, 2, 40] + [4, 5, 30] + [7, 8, 18] + [10, 11, 16] + [12, 13, 23] + [14, 20, 24] + [22, 6, 32]

Table A.2

Catalog of regular graph designs (RGDs) under Series 3.2 with λ₁ = 0 and λ₂ = 1.

v	r	n₁	n₂	α	β	Sets of Shifts
17	8	8	8	2	4	[1, 2, 4] + [5, 8, 1]
29	12	20	8	16	13	[1, 5, 10] + [2, 9, 3] + [8, 4, 7]
41	16	32	8	26	27	[1, 5, 4] + [2, 11, 7] + [14, 3, 16] + [1, 15, 12]
53	20	44	8	39	41	[1, 9, 12] + [2, 15, 13] + [3, 4, 20] + [5, 6, 8] + [16, 1, 18]
65	24	56	8	50	52	[1, 9, 12] + [2, 15, 13] + [3, 4, 20] + [6, 5, 14] + [18, 16, 23] + [1, 32, 29]
77	28	68	8	59	61	[1, 2, 30] + [4, 5, 12] + [6, 7, 24] + [10, 8, 20] + [16, 11, 14] + [23, 22, 26] + [15, 19, 16]
89	32	80	8			[1, 2, 40] + [4, 5, 30] + [10, 11, 16] + [14, 20, 24] + [1, 8, 18] + [12, 13, 23] + [22, 6, 32] + [1, 19, 17]

Table A.3

Catalog of regular graph designs (RGDs) under Series 3.4 with λ₁ = 1 and λ₂ = 2.

v	r	n₁	n₂	α	β	Sets of Shifts
8	4	2	5	0	0	[2, 1, 4]
20	8	14	5	8	12	[1, 6, 9] + [3, 2, 8]
44	16	38	5	32	34	[1, 14, 2] + [3, 7, 12] + [6, 5, 13] + [4, 8, 9]
56	20	50	5	46	46	[2, 16, 10] + [3, 17, 4] + [5, 9, 13] + [8, 25, 12] + [1, 6, 15]
68	24	62	5	56	58	[2, 22, 10] + [13, 4, 14] + [5, 15, 25] + [7, 1, 11] + [26, 3, 30] + [6, 21, 16]

Table A.4

Catalog of regular graph designs (RGDs) under Series 3.5 with λ₁ = 1 and λ₂ = 2.

v	r	n₁	n₂	α	β	Sets of Shifts
14	8	2	11	0	0	[1, 2, 5] + [4, 1, 3]
26	12	14	11	8	6	[1, 3, 8] + [2, 5, 10]
38	16	26	11	17	16	[1, 3, 8] + [2, 5, 10] + [6, 3, 13] + [14, 18, 19]
50	20	38	11	28	31	[1, 8, 3] + [2, 5, 10] + [6, 3, 13] + [14, 4, 19] + [21, 24, 25]
62	24	50	11	40	40	[1, 8, 10] + [2, 15, 11] + [3, 20, 4] + [5, 16, 6] + [7, 13, 12] + [14, 29, 31]
74	28	62	11	50	54	[1, 8, 10] + [2, 15, 11] + [3, 20, 4] + [6, 16, 13] + [14, 7, 25] + [5, 31, 12] + [33, 34, 37]
86	32	74	11	68	64	[1, 8, 10] + [2, 15, 11] + [3, 20, 4] + [6, 16, 13] + [14, 7, 25] + [12, 31, 5] + [37, 41, 42] + [30, 33, 39]
98	36	86	11			[1, 10, 30] + [2, 15, 31] + [3, 18, 24] + [16, 4, 29] + [5, 34, 12] + [6, 19, 13] + [8, 14, 23] + [28, 27, 36] + [9, 26, 44]

Table A.5

Catalog of regular graph designs (RGDs) under Series 3.6 with λ₁ = 1 and λ₂ = 2.

v	r	n₁	n₂	α	β	Sets of Shifts
16	8	6	9	2	2	[1, 2, 4] + [5, 8, 9]
28	12	18	9	8	16	[1, 2, 4] + [5, 8, 9] + [10, 14, 12]
40	16	30	9	20	22	[1, 2, 4] + [5, 8, 9] + [14, 16, 21] + [12, 15, 20]
52	20	42	9	32	34	[1, 10, 13] + [2, 6, 12] + [3, 14, 16] + [4, 5, 21] + [7, 25, 15]
64	24	54	9	46	46	[1, 10, 13] + [2, 6, 12] + [3, 16, 14] + [4, 5, 21] + [15, 7, 25] + [27, 28, 29]
76	28	66	9	56	60	[1, 10, 13] + [2, 6, 12] + [16, 3, 14] + [4, 5, 21] + [7, 15, 25] + [31, 35, 38] + [32, 37, 34]
88	32	78	9	74	70	[1, 20, 19] + [2, 16, 26] + [4, 13, 30] + [3, 5, 28] + [6, 23, 9] + [14, 10, 25] + [11, 31, 37] + [7, 15, 12]
100	36	90	9			[1, 20, 19] + [2, 16, 26] + [4, 13, 30] + [5, 28, 3] + [6, 23, 9] + [14, 10, 25] + [8, 37, 11] + [7, 15, 12] + [41, 46, 50]

Table A.6

Catalog of regular graph designs (RGDs) under Series 3.8 with λ₁ = 1 and λ₂ = 2.

v	r	n₁	n₂	α	β	Sets of Shifts
18	8	10	7	3	6	[1, 2, 4] + [5, 8, 9]
30	12	22	7	16	19	[1, 2, 15] + [4, 5, 14] + [6, 8, 10]
42	16	34	7	26	30	[1, 5, 7] + [3, 11, 9] + [8, 2, 16] + [4, 17, 15]
54	20	46	7	38	40	[1, 5, 7] + [3, 11, 9] + [2, 8, 16] + [4, 17, 15] + [19, 25, 27]
66	24	58	7	52	50	[1, 12, 5] + [2, 27, 6] + [3, 7, 14] + [4, 19, 11] + [8, 20, 22] + [16, 9, 26]
78	28	70	7	62	66	[12, 1, 5] + [4, 19, 11] + [3, 7, 14] + [16, 9, 26] + [2, 29, 8] + [15, 17, 28] + [22, 20, 38]
90	32	82	7	78	76	[1, 15, 17] + [2, 20, 23] + [3, 4, 37] + [13, 5, 21] + [10, 14, 11] + [8, 19, 9] + [6, 34, 38] + [12, 30, 31]

Table A.7

Catalog of regular graph designs (RGDs) under Series 3.9 with λ₁ = 1 and λ₂ = 2.

APPENDIX B

v	Reference		r	λ₁, λ₂	E1	E2	Overall E
8	^b	SR7	4	0, 2	0.75	0.86	0.84
	^b	SR9	6	2, 3	0.83	0.87	0.85
	^d	B6	4				0.85
	^aSer3.5	(0, 2, 3, 7)	4	1, 2	0.8083	0.8661	0.8487
9	^aSer3.1	[1, 2, 4]	4	1, 2	0.8041	0.8616	0.8319
	^b	R8	4	3, 1	0.94	0.77	0.80
	^b	LS1	4	1, 2	0.80	0.87	0.83
	^d	B12	4				0.83
10	^b	S17	4	4, 1	1	0.77	0.79
	^b	T1	2	1, 0	0.83	0.71	0.79
	^b	T2	4	2, 0	0.83	0.71	0.79
	^b	T12	4	1, 2	0.80	0.87	0.82
	^c	T33	4				0.82
	^aSer3.7		4	1, 2	0.7981	0.8646	0.82
11	^aSer3.3		4	1, 2	0.8029	0.86979	0.816
	^d	A25	4				0.817
	^b	SR20	3	0, 1	0.75	0.82	0.80
	^b	R15	4	2, 1	0.88	0.81	0.81
	^c^,^d	R109/B36	4				0.813
	^aDesign 1		4	1, 2	0.8077	0.875	0.8134
14	^b	R24	4	0, 1	0.75	0.81	0.80
	^aSer3.6		8	1, 2	0.7801	0.8101	0.8053
	^aDesign 2		4	0, 1	0.75	0.8077	0.8029
15	^aSer3.1		8	1, 2	0.7776	0.8087	0.7996
	^aDesign 3		4	0, 1	0.7452	0.8073	0.7978
	^b	R27	4	0, 1	0.75	0.80	0.80
	^b	R31	8	1, 2	0.78	0.81	0.80
16	^b	SR40	4	0, 1	0.75	0.80	0.79
	^b	R36	6	2, 1	0.83	0.78	0.79
	^b	LS12	7	2, 1	0.82	0.78	0.79
	^b	LS15	8	2, 1	0.81	0.78	0.80
	^aSer3.8		8	1, 2	0.7778	0.8077	0.7954
17	^b	C5	8	1, 2	0.78	0.81	0.79
	^aSer3.2		4	0, 1	0.7356	0.7969	0.7806
	^aSer3.5		8	1, 2	0.775	0.8073	0.7908
18	^b	S62	8	8, 1	1	0.72	0.73
18	^aSer3.9		8	1, 2	0.7734	0.8071	0.7869
20	^b	SR52	5	0, 1	0.75	0.79	0.78
20	^aSer3.5		8	1, 2	0.7744	0.8096	0.7834
21	^aSer3.1		12	1, 2	0.7692	0.7894	0.7853
22	^b	S82	10	10, 1	1	0.71	0.72
22	^aSer3.7		8	1, 2	0.7775	.8099	0.7819
23	^aSer3.3		8	1, 2	0.7787	0.8112	0.7816
24	^b	R46	7	0, 1	0.75	0.78	0.78
24	^b	R49	10	2, 1	0.80	0.78	0.78
26	^aSer3.5		12	1, 2	0.7683	0.7875	0.7766
26	^b	R53	8	0, 1	0.75	0.78	0.78
27	^aDesign 4		8	0, 1	0.7487	0.7799	0.7774
	^b	R54	8	0, 1	0.75	0.78	0.78
	^aSer3.1		12	1, 2	0.7688	0.788	0.786
28	^aSer3.8		12	1, 2	0.7648	0.7903	0.7731
	^b	SR68	7	0, 1	0.75	0.78	0.77
	^b	R55	8	0, 1	0.75	0.78	0.77
	^b	R56	10	2, 1	0.80	0.77	0.78
29	^aSer3.2		8	0, 1	0.7462	0.7773	0.7727
29	^aSer3.4		12	1, 2	0.7693	0.7881	0.7746
30	^aSer3.9		12	1, 2	0.7679	0.7899	0.7731
30	^b	R57	10	2, 1	0.80	0.77	0.78
32	^b	SR74	8	0, 1	0.75	0.77	0.77
33	^aSer3.1		16	1, 2	0.7641	0.7791	0.7715
38	^aSer3.6		16	1, 2	0.7632	0.7782	0.7676
39	^aSer3.1		16	1, 2	0.7623	0.7789	0.7666
40	^aSer3.8		16	1, 2	0.7627	0.7786	0.7664
	^b	S1.12	4	1, 0	0.77	0.71	0.73
	^b	S1.13	8	2, 0	0.77	0.71	0.73
41	^aSer3.2		12	0, 1	0.7483	0.7686	0.7665
41	^aSer3.4		16	1, 2	0.7641	0.7796	0.7671
42	^aSer3.9		16	1, 2	0.7633	0.7799	0.7661
44	^aSer3.5		16	1, 2	0.7640	0.7799	0.7658
45	^aSer3.1		28	1, 2	0.7586	0.7674	0.7666
50	^aSer3.6		20	1, 2	0.7606	0.7735	0.7635
51	^aSer3.1		20	1, 2	0.7613	0.7736	0.7637
52	^aSer3.8		20	1, 2	0.7606	0.7733	0.7629
53	^aSer3.2		16	0, 1	0.7491	0.7643	0.7631
53	^aSer3.4		20	1, 2	0.7617	0.7744	0.7636
54	^aSer3.9		20	1, 2	0.7611	0.7737	0.7627
56	^aSer3.5		20	1, 2	0.7619	0.7742	0.7630
62	^aSer3.6		24	1, 2	0.7591	0.7694	0.7610
63	^aSer3.1		24	1, 2	0.7594	0.7699	0.7611
64	^aSer3.8		24	1, 2	0.7594	0.7697	0.7609
65	^aSer3.2		20	0, 1	0.7494	0.7618	0.7611
65	^aSer3.4		24	1, 2	0.7597	0.7702	0.761
66	^aSer3.9		24
68	^aSer3.5		24	1, 2	0.7597	0.7702	0.7605
74	^aSer3.6		28	1, 2	0.7577	0.767	0.7591
76	^aSer3.8		28	1, 2	0.7579	0.7672	0.7591
77	^aSer3.2		24	0, 1	0.7496	0.7598	0.7593
77	^aSer3.4		28	1, 2	0.7581	0.7671	0.759
78	^aSer3.9		28	1, 2	0.7582	0.7674	0.759

a

Proposed designs of this article.

b

Designs in Bose et al. [18].

c

Designs in Clatworthy [11].

d

Designs in John et al. [19].

REFERENCES

1.F. Yates, Ann. Eugen., Vol. 7, 1936, pp. 121-140.

2.R.C. Bose and K.R. Nair, Sankhya Indian J. Stat., Vol. 4, 1939, pp. 337-372. https://www.jstor.org/stable/40383923

3.R.C. Bose, Ann. Eugen., Vol. 9, 1939, pp. 353-399.

4.R.C. Bose and T. Shimamoto, J. Am. Stat. Assoc., Vol. 47, 1952, pp. 151-184.

5.R.C. Bose, J. Math., Vol. 13, 1963, pp. 389-419.

6.J.A. John and T.J. Mitchell, J. R. Stat. Soc. Ser. B, Vol. 39, 1977, pp. 39-43.

7.D.L. Kreher, G.F. Royle, and W.D. Wallis, Discrete Math., Vol. 156, 1996, pp. 269-275.

8.F.R. Barandagh, A.R. Barghi, B. Pejman, and M.R. Parsa, Gen. Math. Notes, Vol. 24, 2014, pp. 70-77. https://www.emis.de/journals/GMN/volumes/all_volumes-page2_2014.html

9.S.A. Cakiroglu, Stat. Comput., Vol. 28, 2018, pp. 103-112.

10.K.R. Nair and C.R. Rao, Sci. Cult., Vol. 7, 1942, pp. 568-569.

11.W.H. Clatworthy, Tables of Two Associate Class Partially Balanced Incomplete Block Designs, US Department of Commerce, 1973.

12.C.S. Cheng and C.F. Wu, Biometrika, Vol. 68, 1981, pp. 493-500.

13.W.D. Wallis, J. Stat. Plan. Inference, Vol. 51, 1996, pp. 273-281.

14.V. Kumar, J. Indian Soc. Agric. Stat., Vol. 61, 2007, pp. 38-41.

15.H.B. Waliker, B.D. Acharya, H.S. Ramane, H.G. Shekharappa, and S. Arumugam, AKCE Int. J. Graphs Comb., Vol. 4, 2007, pp. 223-232.

16.F. Yasmin, R. Ahmed, and M. Akhtar, Commun. Stat. Simul. Comput., Vol. 44, 2015, pp. 525-532.

17.I. Iqbal, Construction of Experimental Designs Using Cyclic Shifts, University of Kent, Canterbury, UK, 1991. Ph.D. Thesis

18.R.C. Bose, W.H. Clatworthy, and S.S. Shrikhande, Tables of Partially Balanced Designs with Two Associate Classes, A. N. C. State College Publication, Raleigh, NC, USA, 1954. North Carolina Agricultural Experiment Station Technical Bulletin No. 107

19.J.A. John, F.W. Wolock, and H.A. David, Cyclic Designs, US Department of Commerce, 1972.

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Journal: Journal of Statistical Theory and Applications
Volume-Issue: 19 - 2
Pages: 314 - 324
Publication Date: 2020/07/06
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.200423.001 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

ris enw bib

TY  - JOUR
AU  - Rashid Ahmed
AU  - Farrukh Shehzad
AU  - Muhammad Jamil
AU  - H. M. Kashif Rasheed
PY  - 2020
DA  - 2020/07/06
TI  - Construction of Some Circular Regular Graph Designs in Blocks of Size Four Using Cyclic Shifts
JO  - Journal of Statistical Theory and Applications
SP  - 314
EP  - 324
VL  - 19
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200423.001
DO  - 10.2991/jsta.d.200423.001
ID  - Ahmed2020
ER  -

download .riscopy to clipboard