CMC CMC CMC Computers, Materials & Continua 1526-1506 1526-1492 Tech Science Press USA 15790 10.32604/cmc.2021.015790 Article On Network Designs with Coding Error Detection and Correction Application On Network Designs with Coding Error Detection and Correction Application On Network Designs with Coding Error Detection and Correction Application Higazy Mahmoud 12 m.higazy@tu.edu.sa Nofal Taher A. 1 Department of Mathematics and Statistics, College of Science, Taif University, Taif, 21944, Saudi Arabia Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, 32952, Egypt *Corresponding Author: Mahmoud Higazy. Email: m.higazy@tu.edu.sa 16 01 2021 67 3 3401 3418 07 12 2020 07 01 2021 © 2021 Higazy and Nofal 2021 Higazy and Nofal This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The detection of error and its correction is an important area of mathematics that is vastly constructed in all communication systems. Furthermore, combinatorial design theory has several applications like detecting or correcting errors in communication systems. Network (graph) designs (GDs) are introduced as a generalization of the symmetric balanced incomplete block designs (BIBDs) that are utilized directly in the above mentioned application. The networks (graphs) have been represented by vectors whose entries are the labels of the vertices related to the lengths of edges linked to it. Here, a general method is proposed and applied to construct new networks designs. This method of networks representation has simplified the method of constructing the network designs. In this paper, a novel representation of networks is introduced and used as a technique of constructing the group generated network designs of the complete bipartite networks and certain circulants. A technique of constructing the group generated network designs of the circulants is given with group generated graph designs (GDs) of certain circulants. In addition, the GDs are transformed into an incidence matrices, the rows and the columns of these matrices can be both viewed as a binary nonlinear code. A novel coding error detection and correction application is proposed and examined.

Network decomposition network designs network edge covering circulant graphs
Introduction

Graph (Network) designs are introduced as a generalization of symmetric balanced incomplete block designs (BIBDs) (see, e.g., [1,2]) which are decompositions of complete graphs (networks) to subgraphs (subnetworks) satisfying certain conditions (see ). There are several research papers on the subject of graph decompositions; for more details see . Through the paper we use the word (graph) to mean (network).

As defined in , a symmetric graph design, or SGD, with parameters (n,G,λ;F), where n, λ are positive integers and G and F are graphs with n vertices, is a set {G1,,Gn} of spanning subgraphs of the complete graph Kn such that

Gi G for i=1,,n;

any edge of Kn is contained in exactly λ subgraphs Gi, and

GiGjF for i,j=1,,n, ij.

In , Dalibor Fronček and Alex Rosa determined all graphs F and all orders for which there exists an (n,G,λ;F)-SGD where GFn-12,3, the friendship graph on n vertices.

In this paper, a generalization of symmetric BIBDs is investigated and we introduce a new graph representation that will help in constructing new graph designs (GDs).

Definition 1.1 Let H be a r-regular Cayley graph of order n and B be a non-empty set of spanning subgraphs of H. A(H,B,λ;F)-GD (Graph Design, GD) is a collection Σ={G0,G1,,Gs} of spanning subgraphs of H such that

all graphs G in B have the same size e=|E(G)|r,

any graph of Σ is isomorphic to one graph of B,

every edge of H belongs to exactly λ elements of Σ,

for any two different subgraphs Gi and Gj of Σ, we have GiGjF.

If HKn,n, λ=2, |G|<n and |F|=1 or 0, then the (Kn,n,{G},2;F)-GD is equivalent to the sub-orthogonal double covers (SODCs) of the complete bipartite graph by G. SODC’s have been studied by many authors (for SODCs of Kn, n by G, see  and for SODCs of Kn by G, see, . The (H,{G},2;K2)-GD with |G|=r is equivalent to the orthogonal double covers (ODCs) of Cayley graphs which have been studied in . Also, the (Kn,n,{G},2;K2)-GD with |G|=n is equivalent to ODCs of Kn, n by G that have been investigated by many authors (see, e.g., ). Studying the case when HKn,n, 2$]]>λ>2, |G|=n and FK2 is equivalent to studying the mutually orthogonal graph squares which have been studied by many authors (see, e.g., [7,1619]) and for more details see the survey . Since SODCs and ODCs can be considered as graph designs, its construction tools can be used to construct new graph designs as will be done in this work. Here, all graphs are assumed to be finite, simple and with non-empty edge set. We use the usual notations: n={0,1,,n-1} for the group of all residual classes modulo n, Ø for the empty set, Kn, n for the complete bipartite graphs, Kn for the complete graph, Pn+1 for the path graph with n edges, Sn for the star of size n, En the empty graph of order n, the circulant graph H=Circ(n,A) is defined by V(H)=n and E(H)={(i,i+l):in,lA}, see . In our current study, we concentrate on the case when HKn,n or Circ(n,A) and FK2 or FEn. Note that, if |G|<n, then |Σ| > n. From now on, all addition and subtraction shall be done modulo n. The vertices of Kn, n shall be labeled by the elements of n×2. Namely, for (v,i)n×2 we shall write vi for the corresponding vertex and define {ui,vj}E(Kn,n) if and only if ij, for all u,vn and i,j2. To avoid ambiguity, the edge {u0,v1} shall be written as (u, v). All designs can be represented by a corresponding incidence matrix . Following the method produced in , the incidence matrices can be used in coding error detection and corrections. Here, the suggested codes are not linear codes. The arrangement of our paper is as follows: In Section 2, a new representation of graphs is introduced. In Section 3, a technique of constructing the group generated graph designs of Kn, n is studied. In Section 4, detection of error and its correction is suggested as an application of the codes generated by the constructed graph designs. In Section 5, we construct new group generated graph designs of Kn, n. In Section 6, a technique of constructing the group generated graph designs of the circulants is given with group generated graph designs of certain circulants. The conclusion shall be in Section 7. New Representation of Graphs In this section, we introduce a new representation of graphs following the method that has been introduced in . In , the graphs have been represented by a vector whose entries are the labels of the vertices related to the lengths of edges linked to it. This method of graph representation has simplified the method of constructing the graph designs. Here, a general method is proposed and applied to construct new graph designs. Let G be a spanning subgraph of H and let αn. Then the graph G with E(G+α)={(u+α,v+α):(u,v)E(G)} is called the α-translate of G. The length of an edge e=(u,v)E(G) is defined by l(e) = vu. For any subgraph G of Kn, n, let (G)={y-x:(x,y)E(G)} be the multiset containing the length of every edge in G. For any two subgraphs G1 and G2 of H, let D(G1,G2)={u-x:(x,y)E(G1),(u,v)E(G2),y-x=v-u} be the multiset containing the distance of every pair of equal length edges in G1 and G2. Note that the distance set D(G, G) means the set of distances between the different edges in G which have the same lengths. For any collection of graph Ω={Gi:0ik-1}, we define rd-matrix as a k×k matrix whose entries are rdΩ(i,j)=D(Gi,Gj) for 0ijk-1. Let G be a graph of order n and its vertices are the elements of n, G can be represented by a map ψ(G) from n to its power set (i.e., ψ(G):nP(n)) where for all in, ψi(G)=Ain such that for all aAi, the edge (a,a+i)E(G). ψ(G) can be written in the form of n-tuple where ψi(G)={un:(u,u+i)E(G)} for all 0in-1 (a vector whose ith entry is a set of vertices, from n, incident to the edges with length equal i). Then the following are clear. ψ(Kn,n)=(n,n,n,,n),ψ(Kn)=(Ø,n,n,,n) and for all in, |ψi(Sn)|=1 and ψi(En)=Ø. Let H=Circ(n,A=A+-A+) where A+=A{1,2,,n/2}. Then ψi(H)=ψ-i(H)={nfor all iA,Øotherwise. Let G and H be two spanning subgraphs of Kn, n, G and H are said to be orthogonal if they share at most one edge (i.e., |E(G)E(H)|1), see [8,10] or . Then the collection Ω is mutually orthogonal if and only if all cells of rdΩ matrix are sets. For H = r-regular Circ(n,A), the existence of (H,B,λ;K2)-GD immediately implies the following two necessary conditions that is recorded as Lemma 2.1 Let Σ={G0,G1,,Gs} be a (H,B,λ;K2)-GD and e is the size of any element of B. Then {λnr/2=0mod{e}sλn. Proof. From Definition 1.1 of the (H,B,λ;K2)-GD, we have |E(G1)|+|E(G2)|++|E(Gs)|=λ|E(H)|=λnr/2. Since all elements of Σ are isomorphic to one element of B and all elements of B have the same size e, this implies that λnr/2=se. Also, we have en by Definition 1.1, which imply that sλn. Group Generated Graph Designs of <inline-formula id="ieqn-87"><alternatives><inline-graphic xlink:href="ieqn-87.png"/><tex-math id="tex-ieqn-87"><![CDATA[$\boldsymbol{K}_{{\boldsymbol{n}}, {\boldsymbol{n}}}$]]></tex-math><mml:math id="mml-ieqn-87"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> Definition 3.1 Let Ω={G0,G1,,Gk) be a collection of spanning subgraphs of Kn, n. We call Ω a (Kn,n,B,λ;K2)-GD generator if it satisfies the following conditions: Every element of n appears exactly λ times in the sum of the multisets L(Gi),i=0,1,2,,k-1. For all pairs i, j with 0ijk-1, the cells of the rdΩ matrix are sets, that is D(Gi, Gj) are all sets. The elements of the generator Ω are called (Kn,n,B,λ;K2)-GD pre-starters graphs. Theorem 3.2 Let Ω={G0,G1,,Gk) be a (Kn,n,B,λ;K2)-GD generator. Then for all 0ik-1, the collection of all the translates of Gi+α for all αn, forms a (Kn,n,B,λ;K2)-GD by B. Proof. It is clear that the collection of all translates covers every edge of Kn, n exactly λ times. Now, It is to show that the collection of all translates are mutually orthogonal, that is any two graphs of the collection of all translates share at most one edge. Consider two translates Gi+α and Gj+β where α,βn and assume that they share two edges e1 = (x, y) with length l1 = yx and e2 = (u, v) with length l2 = vu. Then the two edges (x-α,y-α), (u-α,v-α)Gi with lengths l1, l2 respectively and (x-β,y-β), (u-β,v-β)Gj with lengths l1, l2 respectively. Then the distance between the two edges with length l1 in Gi and Gj is α-β, and also the distance between the two edges with length l2 in Gi and Gj is α-β and then D(Gi, Gj) is not a set. This is a contradiction of the second condition in the Definition 3.1 of the (Kn,n,B,λ;K2)-GD generator. Consequently, all subgraphs in the collection of all translates of GD-generator are mutually orthogonal, that is a (Kn,n,B,λ;K2)-GD. Lemma 3.3 Let Ω={G0,G1,,Gk-1) be a (Kn,n,B,λ;K2)-GD generator, then the number of pre-starters in Ω isk=λn/e. For all 0ik-1, if dD(Gi,Gi) then -dD(Gi,Gi), For all 0ik-1, if n is even then n/2D(Gi,Gi). Proof. (i) Σ={G0,G1,,Gs)=(Kn,n,B,λ;K2)-GD. Since s = kn then kne=λn2 and hence k=λn/e. (ii) Let D(Gi, Gi) contains ±d. then Gi contains four edges each pair of them has the same length l1 and l2, that is (x, x + l1), (x + d, x + d + l1), (u,u+l2),(u-d,u-d+l2)Gi. Then Gi + d contains (x + d, x + d + l1), (x + 2d, x + 2d + l1), (u + d, u + d + l2), (u, u + l2) which imply that 1$]]>|GiGi+d|>1 which is a contradiction. Hence, for all 0ik-1, if dD(Gi,Gi) then -dD(Gi,Gi).

(iii) For any 0ik-1, let n/2D(Gi,Gi).

So there exist two edges e1 = (x, x + l), e2 = (x + n/2, x + n/2+l) belong to E(Gi) with the same length l and D(e1, e2) = n/2.

Then Gi + n/2 contains also e1 = (x, x + l), e2 = (x + n/2, x + n/2+l) that means 1]]>|GiGi+n/2|>1 which is a contradiction. Hence, for all 0ik-1, if n is even then n/2D(Gi,Gi). Therefore, λn0mod{e} is a necessary condition of the existence of the (Kn,n,B,λ;K2)-GD generator. Lemma 3.4 Let ψ(G)=(A0,A1,,An-1) is a pre-starter of (Kn,n,B,λ;K2)-GD and Max{|Ai|:0in-1}=m. Then 2 \left(\begin{array}{l}m \\ 2 \end{array}\right)&\text{if }n \text{ even}, \\ n \geq 2 \left(\begin{array}{l}m \\ 2 \end{array}\right)+1&\text{if }n \text{ odd}. \end{array} \end{align*}$]]> n>2(m2 )if n even,n2(m2 )+1if n odd. Proof. Case 1. For n is even; for n2(m2 ), then n/2(m2 ) (the number of differences of the edges of length i), then D(G, G) is a multiset set. This is a contradiction, then 2 \left(\begin{array}{l}m \\ 2 \end{array}\right)$]]>n>2(m2 ).

Case 2. For n is odd; for n<2(m2 )+1, then (n-1)/2<(m2 ) (the number of differences of the edges of length i), then D(G, G) is a multiset set. This is a contradiction, then n2(m2 )+1.

Proposition 3.5 Let m2 and n1 be any integers, B is a set of graphs of size e. If there exists a (Km,m,B,λ;K2)-GD generator of Km, m by B, then there exists a (Kmn,mn,B,λ;K2)-GD generator of Kmn, mn by B.

Proof. Here, the element (s,t)m×n is written as st. Let Ω={G0,G1,,Gk-1} be a (Km,m,B,λ;K2)-GD generator of Km, m by B with respect to m that is every edge in Km, m appears λ times in Ω and D(Gp, Gq) for all 0pqk-1, are sets, and k=λm/e.

For all im and 0sk-1, let ψ(Gs)=(A0s,A1s,,A(m-1)s) is a pre-starter graph GsΩ, that is ψi(Gs)=Ais and E(Gs)={(a,a+i):aAis}.

Let the set D(Gp, Gp) = D1 and the set D(Gp, Gq) = D2 and pq.

For all im, and for all j,tn, define Ω*={G0j,G1j,,Gkj} by ψij(Gst)={Ais×{0}ifj=t,Øotherwise.

Then E(Gst)={(a0,a0+it):aAis}. Then every edge in Kmn, mn (its vertices are m×n×2) appears λ times in Ω*. For any two graphs G,HΩ*,D(G,G)=D1×{0}m×n×{0} which is a set and D(G,H)=D2×{0}m×n×{0} which is a set then Ω* is a (Kmn,mn,B,λ;K2)-GD generator of Kmn,mn by B with respect to m×n.

Coding Error Detection and Correction Application

The rows or columns of the incedence matrix of the GDs can be used as binary codes because all of its entries are 0 or 1. Let us define the GD’s Incedence matrix J as follows.

For the (Kn,n,G,λ;K2)-GD, since Kn, n has n2 edges and we have s blocks (GD subgraphs), define J as s×n2 integer matrix where its elements are 0 or 1 and displays the relation between the edges and the blocks where every row corresponds to a block (GD subgraph Gi) and every column corresponds to an edge (ej) in the graph Kn, n. Jij={1if ejGi0,otherwise. GD Incidence Matrix has the following properties:

As the incidence matrix J of a (Kn,n,G,λ;K2)-GD has the following properties.

Every row has n number of 1s,

Every column has λ number of 1s,

Two distinct columns both have 1s in at most 1 rows.

For illustration, the following example is produced.

The blocks of (K3, 3, S3, 2; K2)-GD is constructed as: {G1={00,01,02},G2={10,11,12},G3={20,21,22}, G4={00,10,20},G5={01,11,21},G6={02,12,22}} where ab is an edge between vertex a0 and vertex b1, see Fig. 1. The incedence matrix of this GD is J=G0G1G2G3G4G5 [111000000000111000000000111100100100010010010001001001 ]

(<italic>K</italic><sub>3, 3</sub>, <italic>S</italic><sub>3</sub>, 2; <italic>K</italic><sub>2</sub>)-GD

When a GD is transformed into an incidence matrix, the rows and the columns can be both viewed as a binary nonlinear code. The binary codes formed from the row denoted as Srow and binary codes from the column will be referred as Scolumn. As mentioned previously, by conversion of GD to incidence matrix, the incidence matrix of a GD retains certain properties that are inherited from GD. Using these properties, results can be obtained to evaluate the minimum Hamming distance (number of different bits in two codes) between codes from Srow or Scolumn. Where Srow={111000000,000111000,000000111,100100100,010010010,001001001} and Scolumn={100100,100010,100001,010100,010010,010001,001100,001010,001001}.

The minimum Hamming distance δ(Srow)=4 and δ(Scolumn)=2.

Distance in binary codes detects the number of errors a code can detect or correct . As proved in , we have

a binary code S can be detected up to q errors iff the minimum distance δ is greater or equivalent to q + 1.

a binary code S can be corrected up to q errors iff the minimum distance δ is greater or equivalent to 2q + 1.

Then for our example Srow can detect upto 3 errors and correct upto one error.

Efficiency factor E is the the quality estimation of the design efficiency. The efficiency factor E is a numerical value lies between 0 and 1. The quality of a design is “good” if E is greater than 0.75 The efficiency of the (v,b,r,k,λ)-BIBD design codes  is calculated as E=v(k-1)k(v-1) which can be simplified for our graph design as E=n2(n-1)n(n2-1)=nn+1 (put v = n2, the size of Kn, n and k = n, the size of G) which will be always greater than 0.75 where n is the size of the GD blocks. Then the efficiency of the codes from the GDs are very good and can be safely used in coding processes. For more details about the design efficiency, see . For more applications of networks, see .

To clear the proposed application, we use the above Srow for coding the following words shown in Tab. 1 and assuming that there is a possibility of occurring an error in at most two positions. From the structure of the corresponding GD, the number of ones must be 3 in any code.

Words’ codes
Words Codes
Go 111000000
Stop 000111000
Forward 000000111
Back 100100100
Left 010010010
Right 001001001

If the code 111100001 is received. Since number of ones must be 3, the error is detected. To correct the error, the code with the minimum Hamming distance from the received one can be chosen that is 111000000. Then the message is “go,” and so on.

Graph Designs <inline-formula id="ieqn-203"><alternatives><inline-graphic xlink:href="ieqn-203.png"/><tex-math id="tex-ieqn-203"><![CDATA[$\boldsymbol{(}\boldsymbol{K}_{{\boldsymbol{n}}, {\boldsymbol{n}}}, \boldsymbol{B}, \boldsymbol{\lambda }; \boldsymbol{K}_{\mathbf{2}}\boldsymbol{)}$]]></tex-math><mml:math id="mml-ieqn-203"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mi>λ</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mn>2</mml:mn></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>-GD’s

Here, we use the above representation of graphs to construct (Kn,n,B,λ;K2)-GD for λ{2,3,4} by certain graph classes B.

Graph Designs <inline-formula id="ieqn-206"><alternatives><inline-graphic xlink:href="ieqn-206.png"/><tex-math id="tex-ieqn-206"><![CDATA[$\boldsymbol{(}\boldsymbol{K}_{{\boldsymbol{n}}, {\boldsymbol{n}}}, \boldsymbol{\{}\boldsymbol{C}_{{\boldsymbol{m}}}\boldsymbol{\}}, \mathbf{2}; \boldsymbol{K}_{\mathbf{2}}\boldsymbol{)}$]]></tex-math><mml:math id="mml-ieqn-206"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mstyle mathvariant="bold"><mml:mn>2</mml:mn></mml:mstyle><mml:mo>;</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mn>2</mml:mn></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>-GD’s

Lemma 5.1 Let t1 be a positive integer. There exists (K6t,6t,{C6},2;K2)-GD.

Proof. For n = 6, define Ω={G0,G1} by ψ(G0)=({0,1},{1,5},{1},Ø,{5},Ø)andψ(G1)=(Ø,Ø,{1},{0,2},{2},{0,1})

Then all graphs in Ω are isomorphic to C6 and rdΩ-matrix=[{1,4}{0,3}{0,3}{1,2} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K6,6,{C6},2;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.2 Let t1 be a positive integer. There exists (K10t,10t,{C10},2;K2)-GD.

Proof. For n = 10, define Ω={G0,G1} by ψ(G0)=({0,9},{8},{0},{9},{6},Ø,{8},{6},{5},{5}),ψ(G1)=(Ø,{0},{0},{7},{5},{5,7},{8},{6},{5},{5})

Then all graphs in Ω are isomorphic to C10 and rdΩ-matrix=[{1,9}{0,6,7,9,5,4,2,8}{0,6,7,9,5,4,2,8}{2,8} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K10t,10t,{C10},2;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Graph Designs <inline-formula id="ieqn-223"><alternatives><inline-graphic xlink:href="ieqn-223.png"/><tex-math id="tex-ieqn-223"><![CDATA[$\boldsymbol{(}\boldsymbol{K}_{{\boldsymbol{n}}, {\boldsymbol{n}}}, \boldsymbol{B}, \mathbf{3}; \boldsymbol{K}_{\mathbf{2}}\boldsymbol{)}$]]></tex-math><mml:math id="mml-ieqn-223"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mstyle mathvariant="bold"><mml:mn>3</mml:mn></mml:mstyle><mml:mo>;</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mn>2</mml:mn></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>-GD’s

The existence of (K4,4,{P5},λ;K2)-GD still open for λ3. Nevertheless, we can record the following result as:

Lemma 5.3 For λ3. There is no (K4,4,{P5},λ;K2)-GD generator.

Proof. Let P5 is a spanning subgraph of K4, 4. Then the following vectors and all of its translates are the all possible pre-starter vectors of P5 shown in Tab. 2. By careful inspection, we find that there are no λ2 mutually orthogonal pre-starter vectors inside this collection, then the proof is complete.

All possible pre-starter vectors of <italic>P</italic><sub>5</sub>
({0,1},{0},{1},Ø) ({0,1},{1},{0},Ø) ({1,1},{0},{2},Ø)
({0,1},Ø,{3},{1}) ({0,1},Ø,{2},{2}) ({0,1},{3},{3},Ø)
({0},{0,1},Ø,{1}) ({1},{0,1},Ø,{0}) (Ø,{0,1},{0},{1})
(Ø,{0,1},{1},{0}) ({1},{0,1},Ø,{3}) (Ø,{0,1},{0},{2})
({1},{0},{0,1},Ø) ({0},{1},{0,1},Ø) ({3},{1},{0,1},Ø)
({2},Ø,{0,1},{0}) ({3},Ø,{0,1},{3}) ({2},{2},{0,1},Ø)
({2},{0,1},{2},Ø) (Ø,{0,1},{3},{3}) ({0},{1},Ø,{0,1})
({1},{0},Ø,{0,1}) (Ø,{0},{1},{0,1}) (Ø,{1},{0},{0,1})
(Ø,{3},{1},{0,1}) (Ø,{2},{2},{0,1}) ({3},{3},Ø,{0,1})
({0},{0},{3},{1}) ({0},{3},{3},{0}) ({0},{3},{1},{0})

Proposition 5.4 Let n3 be a positive integer. There exists a (Kn,n,{P4},3;K2)-GD.

Proof. Define Ω={G0,G1,,Gn-1} as follows.

For all i,jn. ψi(Gj)={{0,1}ifi=j,{0}ifi=j+1,Øotherwise.

Then all graphs in Ω are isomorphic to P4 and E(Gj)={(0,i),(i,i),(0,i+1)}, and D(Gi,Gj)={{1}ifi=j,{0,1}ifj=i+1,Øotherwise.

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (Kn,n,{P4},3;K2)-GD generator.

Lemma 5.5 Let t1 be a positive integer. There exists a (K8t,8t,{C4},3;K2)-GD.

Proof. For n = 8, define Ω={G0,G1,G2,G3,G4,G5} as. ψ(G0)=({0,1},{0},Ø,Ø,Ø,Ø,Ø,{1}),ψ(G1)=(Ø,{1},{0,1},{0},Ø,Ø,Ø,Ø),ψ(G2)=(Ø,Ø,Ø,{1},{0,1},{0},Ø,Ø),ψ(G3)=(Ø,Ø,Ø,Ø,Ø,{1},{0,1},{0}),ψ(G4)=({0},{5},Ø,{5},Ø,Ø,{0},Ø),ψ(G5)=(Ø,Ø,{0},Ø,{0},{5},Ø,{5}).

Then all graphs in Ω are isomorphic to C4 and rdΩ-matrix=[{1}{1}Ø{7}{0,7,5}{4}{1}{1}{1}Ø{4,5}{0,7}Ø{1}{1}{1}{4}{0,7,5}{7}Ø{1}{1}{0,7}{4,5}{0,7,5}{4,5}{4}{0,7}ØØ{4}{0,7}{0,7,5}{4,5}ØØ ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K8,8,{C4},3;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.6 Let t1 be a positive integer. There exists (K6t,6t,{P7},3;K2)-GD.

Proof. For n = 6, define Ω={G0,G1,G2} as. ψ(G0)=({0,1},{0,4},Ø,{3},{1},Ø),ψ(G1)=({1},Ø,{0,1},{0,2},Ø,{2}),ψ(G2)=(Ø,{5},{0},Ø,{4,5},{4,0}).

Then all graphs in Ω are isomorphic to P7 and rdΩ-matrix=[{1,4}{1,0,3,5}{1,5,2,3}{1,0,3,5}{1,2}{0,1,2,4}{1,5,2,3}{0,1,2,4}{1,4} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K6,6,{P7},3;K2)-GD generator, Applying Proposition 3.5 completes the proof.

Lemma 5.7 Let t1 be a positive integer. There exists a (K6t,6t,{C4S2},3;K2)-GD.

Proof. For n = 6, define Ω={G0,G1,G2} by ψ(G0)=({0,5},Ø,{0,4},Ø,{4},{5}),ψ(G1)=({1},{0,5},Ø,{3,5},Ø,{3})ψ(G2)=(Ø,{4},{3},{0},{0,5},{5}).

Then all graphs in Ω are isomorphic to C4S2 and rdΩ-matrix=[{4,5}{1,2,4}{3,5,2,1,0}{1,2,4}{2,5}{4,5,3,1,2}{3,5,2,1,0}{4,5,3,1,2}{5} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K6,6,{C4S2},3;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.8 Let t1 be a positive integer. There exists a (K6t,6t,{C6,P4P3P2},3;K2)-GD.

Proof. For n = 6, define Ω={G0,G1,G2} by ψ(G0)=({0,1},Ø,{5},Ø,{0},{1,5}),ψ(G1)=({1},{0,1},Ø,{0},{4},{4}),ψ(G2)=(Ø,{4},{0,1},{0,2},{3},Ø).

Then {G0,G1} are isomorphic to C6, G2 is isomorphic to P4P3P2 and rdΩ-matrix=[{1,4}{1,0,4,3,5}{1,2,3}{1,0,4,3,5}{1}{4,3,0,2,5}{1,2,3}{4,3,0,2,5}{1,2} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K6,6,{C6,P4P3P2},3;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.9 Let t1 be a positive integer. There exists a (K6t,6t,{C6,P52P2},3;K2)-GD.

Proof. For n = 6, define Ω={G0,G1,G2} by ψ(G0)=({4},{1},{0,1},Ø,{0},{4}),ψ(G1)=({0},Ø,Ø,{0,1},{5},{1,5}),ψ(G2)=({5},{1,4},{5},{1},{2},Ø).

Then {G0,G1} are isomorphic to C6, G2 is isomorphic to P52P2 and rdΩ-matrix=[{1}{2,5,3,1}{1,0,3,5,4,2}{2,5,3,1}{1,4}{5,1,0,3}{1,0,3,5,4,2}{5,1,0,3}{1,2} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K6,6,{C6,P52P2},3;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.10 Let t1 be a positive integer and G is the class of the spanning sub-graphs isomorphic to the graph with vertices {a,b,c,d,e,r,s} and the 6 edges {(a,b),(c,b),(c,d),(e,b),(e,d),(r,s)}. There exists a (K6t,6t,{C6,G},3;K2)-GD.

Proof. For n = 6, define Ω={G0,G1,G2} by ψ(G0)=({4},{1},{0,1},Ø,{0},{4}),ψ(G1)=({0},Ø,Ø,{0,1},{5},{1,5}),ψ(G2)=({1},{0,1},{2},{4},{4},Ø).

Then {G0,G1} are isomorphic to C6, G2 is isomorphic to G and rdΩ-matrix=[{0,1}{2,5,3,1}{3,5,0,2,1,4}{2,5,3,1}{1,4}{1,4,3,5}{3,5,0,2,1,4}{4,3,0,2,5}{1} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K6,6,{C6,G},3;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.11 Let t1 be a positive integer. There exists a (K8t,8t,{C6},3;K2)-GD.

Proof. For n = 8, define Ω={G0,G1,G2,G3} as. ψ(G0)=(Ø,Ø,{6},Ø,Ø,{0},{1},{0,1,6}),ψ(G1)=({0,1,6},{0},{6},Ø,Ø,{1},Ø,Ø),ψ(G2)=(Ø,{0},{1},{0,1,6},Ø,Ø,{6},Ø),ψ(G3)=(Ø,{1},Ø,Ø,{0,1,6},{0},{6},Ø).

Then all graphs in Ω are isomorphic to C6 and rdΩ-matrix=[{1,2,3}{0,1}{5,-5}{0,5}{0,1}{1,2,3}{0,5}{1,-1}{5,-5}{0,5}{1,2,3}{0,1}{0,5}{1,-1}{0,1}{1,2,3} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K8,8,{C6},3;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.12 Let t1 be a positive integer and G is a graph containing a cycle C4 in addition to an edge K2such that they share a vertex. There exists (K5t,5t,{C4K2,G},3;K2)-GD.

Proof. For n = 5, define Ω={G0,G1,G2} as: ψ(G0)=({0,3},Ø,{3},{0},{2}),ψ(G1)=({3},{2,3},{2,0},Ø,Ø),ψ(G2)=(Ø,{2},Ø,{3,4},{2,4}).

Then {G0,G1} are isomorphic to C4, G2 is isomorphic to G and rdΩ-matrix=[{3}{3,0,4,2}{3,4,0,2}{3,0,4,2}{1,3}{0,4}{3,4,0,2}{0,4}{1,2} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K5,5,{C4K2,G},3;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.13 Let t1 be a positive integer. There exists a (K5t,5t,{P6},3;K2)-GD.

Proof. For n = 5, define Ω={G0,G1,G2} as: ψ(G0)=({0,3},{3,4},Ø,{0},Ø),ψ(G1)=(Ø,{2},{0,2},Ø,{0,4}),ψ(G2)=({4},Ø,{4},{0,3},{0}).

Then all graphs in Ω are isomorphic to P6 and rdΩ-matrix=[{3,1}{4,3}{4,1,0,3}{4,3}{2,4}{4,2,0,1}{4,1,0,3}{4,2,0,1}{3} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K5,5,{P6},3;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.14 Let t1 be a positive integer. There exists a (K5t,5t,{P6,P42K2},3;K2)-GD.

Proof. For n = 5, define Ω={G0,G1,G2} as. ψ(G0)=({0,1},{0},{1},Ø,{4}),ψ(G1)=(Ø,{3},{1},{0,1},{3}),ψ(G2)=({4},{1},{3},{0},{4}).

Then {G0,G1} are isomorphic to P6, G2 is isomorphic to P42K2 and rdΩ-matrix=[{1}{3,0,4}{4,3,1,2,0}{3,0,4}{1}{3,2,0,4,1}{4,3,1,2,0}{3,2,0,4,1}Ø ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K5,5,{P6,P42K2},3;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Graph Designs <inline-formula id="ieqn-354"><alternatives><inline-graphic xlink:href="ieqn-354.png"/><tex-math id="tex-ieqn-354"><![CDATA[$\boldsymbol{(}\boldsymbol{K}_{{\boldsymbol{n}}, {\boldsymbol{n}}}, \boldsymbol{B}, \mathbf{4}; \boldsymbol{K}_{\mathbf{2}}\boldsymbol{)}$]]></tex-math><mml:math id="mml-ieqn-354"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mstyle mathvariant="bold"><mml:mn>4</mml:mn></mml:mstyle><mml:mo>;</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mn>2</mml:mn></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>-GD’s

Lemma 5.15 Let t1 be a positive integer. There exists a (K5t,5t,{P4P3,P5K2},4;K2)-GD.

Proof. For n = 5, define Ω={G0,G1,G2,G3} as: ψ(G0)=({0,2},Ø,{4},{2,3},Ø),ψ(G1)=({0,1},{2,4},{1},Ø,Ø),ψ(G2)=(Ø,{4,0},Ø,{4},{0,3}),ψ(G3)=(Ø,Ø,{0,2},{2},{0,4}).

Then {G0,G1,G2} are isomorphic to P4P3, G3 is isomorphic to P4P3 and rdΩ-matrix=[{2,1}{0,3,1,4,2}{2,1}{1,3,0,4}{0,3,1,4,2}{1,2}{2,0,3,1}{2,4}{2,1}{2,0,3,1}{1,3}{4,0,2,1,3}{1,3,0,4}{2,4}{4,0,2,1,3}{2,4} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K5,5,{P4P3,P5K2},4;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.16 Let t1 be a positive integer. There exists a (K5t,5t,{P4K2},4;K2)-GD.

Proof. For n = 5, define Ω={G0,G1,G2,G3} as: ψ(G0)=({0,2},Ø,Ø,{1,2},Ø),ψ(G1)=({0,1},Ø,Ø,Ø,{1,3}),ψ(G2)=(Ø,{2,3},Ø,{0,2},Ø),ψ(G3)=(Ø,Ø,{0,2},Ø,{0,4}),ψ(G4)=(Ø,{0,2},{1,2},Ø,Ø).

Then all graphs in Ω are isomorphic to P4K2 and rdΩ-matrix=[{2,1}{0,3,1,4}ØØØ{0,3,1,4}{1,2}Ø{4,2,3,1}ØØØ{1,2}Ø{3,2,0,4}Ø{4,2,3,1}Ø{2,4}{1,4,2,0}ØØ{3,2,0,4}{1,4,2,0}{2,1} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 3.3, then Ω is a (K5,5,{P4K2},4;K2)-GD generator. Applying Proposition 3.5 completes the proof.

Graph Designs <inline-formula id="ieqn-374"><alternatives><inline-graphic xlink:href="ieqn-374.png"/><tex-math id="tex-ieqn-374"><![CDATA[$\boldsymbol{(}\boldsymbol{H}, \boldsymbol{B}, \boldsymbol{\lambda }; \boldsymbol{K}_{\mathbf{2}}\boldsymbol{)}$]]></tex-math><mml:math id="mml-ieqn-374"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>H</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mi>λ</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mn>2</mml:mn></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>-GD’s

Definition 6.1 Let Ω={G0,G1,,Gg) be a collection of spanning subgraphs of H = r-regular Circ(n,A=A+-A+) where A+=A{1,2,,n/2}. We call Ω a (H,B,λ;K2)-GD generator if it satisfies the following conditions:

Every element of A+ appears exactly λ times in the sum of the multisets L(Gi), i=0,1,2,,g-1.

For all pairs i, j with 0ijg-1, the cells of the rdΩ matrix are sets, that is D(Gi, Gj) are all sets.

The elements of the generator Ω are called (H,B,λ;K2)-GD pre-starters graphs.

Theorem 6.2 Let Ω={G0,G1,,Gg) be a (H,B,λ;K2)-GD generator. Then for all 0ig-1, the collection of all the translates of Gi+α for all xn, forms a (H,B,λ;K2)-GD by B.

Proof. It is clear that the collection of all translates covers every edge of H exactly λ times. Now, It is to show that the collection of all translates are mutually orthogonal, that is any two graphs of the collection of all translates share at most one edge. Consider two translates Gi+α and Gj+β where α,βn and assume that they share two edges e1 = (x, y) with length l1 = yx and e2 = (u, v) with length l2 = vu. Then the two edges (x-α,y-α), (u-α,v-α)Gi with lengths l1, l2 respectively and (x-β,y-β), (u-β,v-β)Gj with lengths l1, l2 respectively. Then the distance between the two edges with length l1 in Gi and Gj is α-β, and also the distance between the two edges with length l1 in Gi and Gj is α-β and then D(Gi, Gj) is not a set. This is a contradiction of the second condition in the Definition 6.1 of the (H,B,λ;K2)-GD generator. Consequently, all subgraphs in the collection of all translates of GD-generator are mutually orthogonal, that is a (H,B,λ;K2)-GD.

Lemma 6.3 Let Ω={G0,G1,,Gg-1) be a (H,B,λ;K2)-GD generator, then

the number of pre-starters in Ω isg=λnr/2e,

For all 0ig-1, if dD(Gi,Gi) then-dD(Gi,Gi),

For all 0ig-1, if n is even then n/2D(Gi,Gi).

Proof. (i) Σ={G0,G1,,Gs)=(H,B,λ;K2)-GD. Since the s = gn then gne=λnr/2 and hence g=λr/2e.

(ii) Let D(Gi, Gi) contains ±d then Gi contains four edges each pair of them has the same length l1 and l2, that is (x, x + l1), (x + d, x + d + l1), (u, u + l2), (u-d,u-d+l2)Gi.

Then Gi + d contains (x + d, x + d + l1), (x + 2d, x + 2d + l1), (u + d, u + d + l2), (u, u + l2) which imply that 1$]]>|GiGi+d|>1 which is a contradiction. Hence, for all 0ig-1, if dD(Gi,Gi) then -dD(Gi,Gi) (iii) For any 0ig, let n/2D(Gi,Gi). So there exist two edges e1 = (x, x + l), e2 = (x + n/2, x + n/2+l) belong to E(Gi) with the same length l and D(e1, e2) = n/2. Then Gi + n/2 contains also e1 = (x, x + l), e2 = (x + n/2, x + n/2+l) that means 1$]]>|GiGi+n/2|>1 which is a contradiction. Hence, for all 0ig-1, if n is even then n/2D(Gi,Gi).

Therefore, λr/20mod{e} is a necessary condition of the existence of the (H,B,λ;K2)-GD generator.

Proposition 6.4 Let m2 and n2m+1 be integers and let H = 2m-regular Circ(n,A) where A=A+-A+ where A+=A{1,2,,n/2}={l0,l1,,lm-1}. Then there exists (H,{P4},3;K2)-GD.

Proof. Define Ω={G0,G1,,Gm-1} as:

For all jm and for all im ψli(Gj)={{0,lj+1}ifi=j,{lj}ifi=j+1,Øotherwise.

Then all graphs in Ω are isomorphic to P4 and D(Gi,Gj)={{lj+1}if i=j,{-lj,lj+2-lj}if i=j-1,Øotherwise.

Since every cell of the rdΩ-matrix is a set satisfying Lemma 6.3, then Ω is a (H,{P4},3;K2)-GD generator.

Proposition 6.5 Let n9 be an integer and let H = 8 -regular Circ(n,A) where A=A=A+-A+ where A+=A{1,2,,n/2}={l1,l2,l3,l4}. Then there exists (H,{P5},3;K2)-GD.

Proof. Define Ω={G0,G1,G2} as: ψi(G0)={{0,l1,l4-l1}ifi=l1,{0}ifi=l4,Øotherwise. ψi(G1)={{0,l2,l3-l2}ifi=l2,{0}ifi=l3,Øotherwise. ψi(G2)={{0,l3+l4}ifi=l3,{0,l3}ifi=l4,Øotherwise.

Then all graphs in Ω are isomorphic to P5 and rdΩ-matrix=[{l1,l4-l1,l4-2l1}Ø{0,l3}Ø{1}{0,l3+l4}{0,l3}{0,l3+l4}Ø ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 6.3, then Ω is a (H,{P5},3;K2)-GD generator.

Proposition 6.6 Let n7 be a positive integer and H be 4-regular Circ(n,A) where A=A=A+-A+ where A+=A{1,2,,n/2}={l1,l2} such that {l2,l1-l2,l1-2l2}, {l1,l1+l2,2l1+l2}, {0,-l1,l1+l2,l1,l1-l2,l2}, {l1-l2,l1,0,l2,2l2} and {-l1,0,l1,l1+l2,2l1+l2} are all sets (i.e., all have different elements).Then there exists (H,{P5},4;K2)-GD.

Proof. Define Ω={G0,G1} as ψi(G0)={{0,-l1,l1+l2}ifi=l1,{l1}ifi=l2,Øotherwise. ψi(G1)={{0}ifi=l1{0,l2,l1-l2}ifi=l2,Øotherwise.

Since {l1-l2,l1,0,l2,2l2} and {-l1,0,l1,l1+l2,2l1+l2} are sets then all graphs in Ω are isomorphic to P5 and rdΩ-matrix=[{l2,l1-l2,l1-2l2}{0,-l1,l1+l2,l1,l1-l2,l2}{0,-l1,l1+l2,l1,l1-l2,l2}{l1,l1+l2,2l1+l2} ]

Since every cell of the rdΩ-matrix is a set satisfying Lemma 6.3, then Ω is a (H,{P5},4;K2)-GD generator. For illustration, at n = 7 take l1 = 1 and l2 = 3.

New graph designs
H B λ F H B λ F
K6t, 6t {C6} 2 K2 K5t, 5t {P6,P42K2} 3 K2
K10t, 10t {C10} 2 K2 K6t, 6t {C6,P52P2} 3 K2
K8t, 8t {C4} 3 K2 K6t, 6t {C6,G} 3 K2
K6t, 6t {P7} 3 K2 K8t, 8t {C6} 3 K2
K6t, 6t {C4S2} 3 K2 Kn, n {P4} 3 K2
K6t, 6t {C6,P4P3P2} 3 K2 K5t, 5t {C4K2,G} 3 K2
K6t, 6t {C6,P52P2} 3 K2 K5t, 5t {P6} 3 K2
K8t, 8t {C6} 3 K2 K5t, 5t {P6,P42K2} 3 K2
Kn, n {P4} 3 K2 K5t, 5t {P4P3,P5K2} 3 K2
K5t, 5t {P6} 3 K2 K5t, 5t {P4K2} 3 K2
K5t, 5t {P6,P42K2} 3 K2 H {P4} 3 K2
H {P5} 4 K2 H {P5} 3 K2
Conclusion

In this paper, we have studied the group generated graph designs. A new representation of graphs has been proposed that help in constructing new graph designs (H,B,λ;F)-GD that can be summerized in Tab. 3. Where H is certain circulant graph. In addition, an efficient coding method has been proposed using the constructed graph designs which may open a new door to produce more research in this area. Finally, we can state that the constructed GD’s can be efficiently used to generate a code set.

The authors are thankful of the Taif University. Taif University researchers supporting project number (TURSP-2020/031), Taif University, Taif, Saudi Arabia.

Funding Statement: The authors received financial support from Taif University Researchers Supporting Project Number (TURSP-2020/031), Taif University, Taif, Saudi Arabia.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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