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.
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 [1]). There are several research papers on the subject of graph decompositions; for more details see [3]. Through the paper we use the word (graph) to mean (network).
As defined in [1], 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 K_{n} such that
G_{i}≅G for i=1,…,n;
any edge of K_{n} is contained in exactly λ subgraphs G_{i}, and
Gi∩Gj≅F for i,j=1,…,n, i≠j.
In [4], Dalibor Fronček and Alex Rosa determined all graphs F and all orders for which there exists an (n,G,λ;F)-SGD where G≅Fn-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 1.1Let 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 collectionΣ={G0,G1,…,Gs}of spanning subgraphs of H such that
all graphs G in B have the same sizee=|E(G)|≤r,
any graph ofΣis isomorphic to one graph of B,
every edge of H belongs to exactlyλelements ofΣ,
for any two different subgraphs G_{i}and G_{j}ofΣ, we haveGi∩Gj≅F.
If H≅Kn,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 K_{n, n} by G, see [5–7] and for SODCs of K_{n} by G, see, [8–10]. 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 [11]. Also, the (Kn,n,{G},2;K2)-GD with |G|=n is equivalent to ODCs of K_{n, n} by G that have been investigated by many authors (see, e.g., [12–15]). Studying the case when H≅Kn,n, 2$]]>λ>2, |G|=n and F≅K2 is equivalent to studying the mutually orthogonal graph squares which have been studied by many authors (see, e.g., [7,16–19]) and for more details see the survey [20]. 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: ℤn={0,1,…,n-1} for the group of all residual classes modulo n, Ø for the empty set, K_{n, n} for the complete bipartite graphs, K_{n} for the complete graph, P_{n+1} for the path graph with n edges, S_{n} for the star of size n, E_{n} the empty graph of order n, the circulant graph H=Circ(ℤn,A) is defined by V(H)=ℤn and E(H)={(i,i+l):i∈ℤn,l∈A}, see [21].
In our current study, we concentrate on the case when H≅Kn,n or Circ(ℤn,A) and F≅K2 or F≅En. Note that, if |G|<n, then |Σ| > n.
From now on, all addition and subtraction shall be done modulo n.
The vertices of K_{n, n} shall be labeled by the elements of ℤn×ℤ2. Namely, for (v,i)∈ℤn×ℤ2 we shall write v_{i} for the corresponding vertex and define {ui,vj}∈E(Kn,n) if and only if i≠j, for all u,v∈ℤn and i,j∈ℤ2. To avoid ambiguity, the edge {u0,v1} shall be written as (u, v).
All designs can be represented by a corresponding incidence matrix [22]. Following the method produced in [23], 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 K_{n, 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 K_{n, 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 [13]. In [13], 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 α∈ℤn. 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) = v − u.
For any subgraph G of K_{n, n}, let (G)={y-x:(x,y)∈E(G)}
be the multiset containing the length of every edge in G. For any two subgraphs G_{1} and G_{2} 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 G_{1} and G_{2}. 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:0≤i≤k-1}, we define rd-matrix as a k×k matrix whose entries are rdΩ(i,j)=D(Gi,Gj) for 0≤i≤j≤k-1.
Let G be a graph of order n and its vertices are the elements of ℤn, G can be represented by a map ψ(G) from ℤn to its power set (i.e., ψ(G):ℤn→P(ℤn)) where for all i∈ℤn, ψi(G)=Ai⊆ℤn such that for all a∈Ai, the edge (a,a+i)∈E(G). ψ(G) can be written in the form of n-tuple where
ψi(G)={u∈ℤn:(u,u+i)∈E(G)}
for all 0≤i≤n-1 (a vector whose ith entry is a set of vertices, from ℤn, incident to the edges with length equal i). Then the following are clear.
ψ(Kn,n)=(ℤn,ℤn,ℤn,…,ℤn),ψ(Kn)=(Ø,ℤn,ℤn,…,ℤn)
and for all i∈ℤn, |ψi(Sn)|=1 and ψi(En)=Ø.
Let H=Circ(ℤn,A=A+∪-A+) where A+=A∩{1,2,…,⌊n/2⌋}. Then
ψi(H)=ψ-i(H)={ℤnfor all i∈A,Øotherwise.
Let G and H be two spanning subgraphs of K_{n, 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 [24]. Then the collection Ω is mutually orthogonal if and only if all cells of rdΩ matrix are sets.
For H = r-regular Circ(ℤn,A), the existence of (H,B,λ;K2)-GD immediately implies the following two necessary conditions that is recorded as
Lemma 2.1LetΣ={G0,G1,…,Gs}be a(H,B,λ;K2)-GD and e is the size of any element ofB. Then{λnr/2=0mod{e}s≥λn.
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 e≤n by Definition 1.1, which imply that s≥λn.
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 3.1LetΩ={G0,G1,…,Gk)be a collection of spanning subgraphs of K_{n, n}. We callΩa(Kn,n,B,λ;K2)-GD generator if it satisfies the following conditions:
Every element ofℤnappears exactlyλtimes in the sum of the multisets
L(Gi),i=0,1,2,…,k-1.
For all pairs i, j with0≤i≤j≤k-1, the cells of therdΩmatrix are sets, that is D(G_{i}, G_{j}) are all sets.
The elements of the generator Ω are called (Kn,n,B,λ;K2)-GD pre-starters graphs.
Theorem 3.2LetΩ={G0,G1,…,Gk)be a(Kn,n,B,λ;K2)-GD generator. Then for all0≤i≤k-1, the collection of all the translates ofGi+αfor allα∈ℤn, forms a(Kn,n,B,λ;K2)-GD by B.
Proof. It is clear that the collection of all translates covers every edge of K_{n, 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,β∈ℤn and assume that they share two edges e_{1} = (x, y) with length l_{1} = y − x and e_{2} = (u, v) with length l_{2} = v − u. Then the two edges (x-α,y-α), (u-α,v-α)∈Gi with lengths l_{1}, l_{2} respectively and (x-β,y-β), (u-β,v-β)∈Gj with lengths l_{1}, l_{2} respectively. Then the distance between the two edges with length l_{1} in G_{i} and G_{j} is α-β, and also the distance between the two edges with length l_{2} in G_{i} and G_{j} is α-β and then D(G_{i}, G_{j}) 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 3.3LetΩ={G0,G1,…,Gk-1)be a(Kn,n,B,λ;K2)-GD generator, then
the number of pre-starters inΩisk=λn/e.
For all0≤i≤k-1, ifd∈D(Gi,Gi)then-d∉D(Gi,Gi),
For all0≤i≤k-1, if n is even thenn/2∉D(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(G_{i}, G_{i}) contains ±d. then G_{i} contains four edges each pair of them has the same length l_{1} and l_{2}, that is (x, x + l_{1}), (x + d, x + d + l_{1}), (u,u+l2),(u-d,u-d+l2)∈Gi.
Then G_{i} + d contains (x + d, x + d + l_{1}), (x + 2d, x + 2d + l_{1}), (u + d, u + d + l_{2}), (u, u + l_{2}) which imply that 1$]]>|Gi∩Gi+d|>1 which is a contradiction. Hence, for all 0≤i≤k-1, if d∈D(Gi,Gi) then -d∉D(Gi,Gi).
(iii) For any 0≤i≤k-1, let n/2∈D(Gi,Gi).
So there exist two edges e_{1} = (x, x + l), e_{2} = (x + n/2, x + n/2+l) belong to E(G_{i}) with the same length l and D(e_{1}, e_{2}) = n/2.
Then G_{i} + n/2 contains also e_{1} = (x, x + l), e_{2} = (x + n/2, x + n/2+l) that means 1$]]>|Gi∩Gi+n/2|>1 which is a contradiction. Hence, for all 0≤i≤k-1, if n is even then n/2∉D(Gi,Gi).
Therefore, λn≡0mod{e} is a necessary condition of the existence of the (Kn,n,B,λ;K2)-GD generator.
Lemma 3.4Letψ(G)=(A0,A1,…,An-1)is a pre-starter of(Kn,n,B,λ;K2)-GD andMax{|Ai|:0≤i≤n-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,n≥2(m2)+1if n odd.
Proof. Case 1. For n is even; for n≤2(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 n≥2(m2)+1.
Proposition 3.5Letm≥2 and n≥1be any integers, B is a set of graphs of size e. If there exists a(Km,m,B,λ;K2)-GD generator of K_{m, m}by B, then there exists a(Kmn,mn,B,λ;K2)-GD generator of K_{mn, mn}by B.
Proof. Here, the element (s,t)∈ℤm×ℤn is written as st. Let Ω={G0,G1,…,Gk-1} be a (Km,m,B,λ;K2)-GD generator of K_{m, m} by B with respect to ℤm that is every edge in K_{m, m} appears λ times in Ω and D(G^{p}, G^{q}) for all 0≤p≤q≤k-1, are sets, and k=λm/e.
For all i∈ℤm and 0≤s≤k-1, let ψ(Gs)=(A0s,A1s,…,A(m-1)s) is a pre-starter graph Gs∈Ω, that is ψi(Gs)=Ais and E(Gs)={(a,a+i):a∈Ais}.
Let the set D(G_{p}, G_{p}) = D_{1} and the set D(G_{p}, G_{q}) = D_{2} and p≠q.
For all i∈ℤm, and for all j,t∈ℤn, define Ω*={G0j,G1j,…,Gkj} by
ψij(Gst)={Ais×{0}ifj=t,Øotherwise.
Then E(Gst)={(a0,a0+it):a∈Ais}. Then every edge in K_{mn, mn} (its vertices are ℤm×ℤn×ℤ2) appears λ times in Ω*. For any two graphs G,H∈Ω*,D(G,G)=D1×{0}⊆ℤm×ℤn×{0} which is a set and D(G,H)=D2×{0}⊆ℤm×ℤn×{0} which is a set then Ω* is a (Kmn,mn,B,λ;K2)-GD generator of Kmn,mn by B with respect to ℤm×ℤn.
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 K_{n, n} has n^{2} 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 G_{i}) and every column corresponds to an edge (e_{j}) in the graph K_{n, n}.
Jij={1if ej∈Gi0,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 (K_{3, 3}, S_{3}, 2; K_{2})-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 a_{0} and vertex b_{1}, see Fig. 1. The incedence matrix of this GD is
J=G0G1G2G3G4G5[111000000000111000000000111100100100010010010001001001]
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 [25]. As proved in [26], 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 [27] 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 = n^{2}, the size of K_{n, 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 [27]. For more applications of networks, see [28–30].
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.
Lemma 5.1Lett≥1 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 C_{6} 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 5.2Lett≥1be 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 C_{10} 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. ■
The existence of (K4,4,{P5},λ;K2)-GD still open for λ≥3. Nevertheless, we can record the following result as:
Lemma 5.3Forλ≥3. There is no(K4,4,{P5},λ;K2)-GD generator.
Proof. Let P_{5} is a spanning subgraph of K_{4, 4}. Then the following vectors and all of its translates are the all possible pre-starter vectors of P_{5} shown in Tab. 2. By careful inspection, we find that there are no λ≥2 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 5.4Letn≥3be a positive integer. There exists a(Kn,n,{P4},3;K2)-GD.
Proof. Define Ω={G0,G1,…,Gn-1} as follows.
For all i,j∈ℤn.
ψi(Gj)={{0,1}ifi=j,{0}ifi=j+1,Øotherwise.
Then all graphs in Ω are isomorphic to P_{4} 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 5.5Lett≥1be 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 C_{4} 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 5.6Lett≥1be 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 P_{7} 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 5.7Lett≥1be a positive integer. There exists a(K6t,6t,{C4∪S2},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 C4∪S2 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,{C4∪S2},3;K2)-GD generator. Applying Proposition 3.5 completes the proof. ■
Lemma 5.8Lett≥1be a positive integer. There exists a(K6t,6t,{C6,P4∪P3∪P2},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 C_{6}, G_{2} is isomorphic to P4∪P3∪P2 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,P4∪P3∪P2},3;K2)-GD generator. Applying Proposition 3.5 completes the proof.
Lemma 5.9Lett≥1be a positive integer. There exists a(K6t,6t,{C6,P5∪2P2},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 C_{6}, G_{2} is isomorphic to P5∪2P2 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,P5∪2P2},3;K2)-GD generator. Applying Proposition 3.5 completes the proof. ■
Lemma 5.10Lett≥1be 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 C_{6}, G_{2} 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 5.11Lett≥1be 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 C_{6} 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 5.12Lett≥1be a positive integer and G is a graph containing a cycle C_{4}in addition to an edge K_{2}such that they share a vertex. There exists(K5t,5t,{C4∪K2,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 C_{4}, G_{2} 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,{C4∪K2,G},3;K2)-GD generator. Applying Proposition 3.5 completes the proof. ■
Lemma 5.13Lett≥1be 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 P_{6} 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 5.14Lett≥1be a positive integer. There exists a(K5t,5t,{P6,P4∪2K2},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 P_{6}, G_{2} is isomorphic to P4∪2K2 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,P4∪2K2},3;K2)-GD generator. Applying Proposition 3.5 completes the proof. ■
Lemma 5.15Lett≥1be a positive integer. There exists a(K5t,5t,{P4∪P3,P5∪K2},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 P4∪P3, G_{3} is isomorphic to P4∪P3 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,{P4∪P3,P5∪K2},4;K2)-GD generator. Applying Proposition 3.5 completes the proof. ■
Lemma 5.16Lett≥1be a positive integer. There exists a(K5t,5t,{P4∪K2},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 P4∪K2 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,{P4∪K2},4;K2)-GD generator. Applying Proposition 3.5 completes the proof. ■
Definition 6.1LetΩ={G0,G1,…,Gg)be a collection of spanning subgraphs of H = r-regularCirc(ℤn,A=A+∪-A+)whereA+=A∩{1,2,…,⌊n/2⌋}. We callΩa(H,B,λ;K2)-GD generator if it satisfies the following conditions:
Every element of A^{+}appears exactlyλtimes in the sum of the multisets L(G_{i}), i=0,1,2,…,g-1.
For all pairs i, j with0≤i≤j≤g-1, the cells of therdΩmatrix are sets, that is D(G_{i}, G_{j}) are all sets.
The elements of the generator Ω are called (H,B,λ;K2)-GD pre-starters graphs.
Theorem 6.2LetΩ={G0,G1,…,Gg)be a(H,B,λ;K2)-GD generator. Then for all0≤i≤g-1, the collection of all the translates ofGi+αfor allx∈ℤn, 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,β∈ℤn and assume that they share two edges e_{1} = (x, y) with length l_{1} = y − x and e_{2} = (u, v) with length l_{2} = v − u. Then the two edges (x-α,y-α), (u-α,v-α)∈Gi with lengths l_{1}, l_{2} respectively and (x-β,y-β), (u-β,v-β)∈Gj with lengths l_{1}, l_{2} respectively. Then the distance between the two edges with length l_{1} in G_{i} and G_{j} is α-β, and also the distance between the two edges with length l_{1} in G_{i} and G_{j} is α-β and then D(G_{i}, G_{j}) 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 6.3LetΩ={G0,G1,…,Gg-1)be a(H,B,λ;K2)-GD generator, then
the number of pre-starters inΩisg=λnr/2e,
For all0≤i≤g-1, ifd∈D(Gi,Gi)then-d∉D(Gi,Gi),
For all0≤i≤g-1, if n is even thenn/2∉D(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(G_{i}, G_{i}) contains ±d then G_{i} contains four edges each pair of them has the same length l_{1} and l_{2}, that is (x, x + l_{1}), (x + d, x + d + l_{1}), (u, u + l_{2}), (u-d,u-d+l2)∈Gi.
Then G_{i} + d contains (x + d, x + d + l_{1}), (x + 2d, x + 2d + l_{1}), (u + d, u + d + l_{2}), (u, u + l_{2}) which imply that 1$]]>|Gi∩Gi+d|>1 which is a contradiction. Hence, for all 0≤i≤g-1, if d∈D(Gi,Gi) then -d∉D(Gi,Gi)
(iii) For any 0≤i≤g, let n/2∈D(Gi,Gi).
So there exist two edges e_{1} = (x, x + l), e_{2} = (x + n/2, x + n/2+l) belong to E(G_{i}) with the same length l and D(e_{1}, e_{2}) = n/2.
Then G_{i} + n/2 contains also e_{1} = (x, x + l), e_{2} = (x + n/2, x + n/2+l) that means 1$]]>|Gi∩Gi+n/2|>1 which is a contradiction. Hence, for all 0≤i≤g-1, if n is even then n/2∉D(Gi,Gi).
Therefore, λr/2≡0mod{e} is a necessary condition of the existence of the (H,B,λ;K2)-GD generator. ■
Proposition 6.4Letm≥2andn≥2m+1be integers and let H = 2m-regularCirc(ℤn,A)whereA=A+∪-A+whereA+=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 j∈ℤm and for all i∈ℤmψli(Gj)={{0,lj+1}ifi=j,{lj}ifi=j+1,Øotherwise.
Then all graphs in Ω are isomorphic to P_{4} 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 6.5Letn≥9be an integer and let H = 8 -regularCirc(ℤn,A)whereA=A=A+∪-A+whereA+=A∩{1,2,…,⌊n/2⌋}={l1,l2,l3,l4}. Then there exists(H,{P5},3;K2)-GD.
Then all graphs in Ω are isomorphic to P_{5} 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 6.6Letn≥7be a positive integer and H be 4-regularCirc(ℤn,A)whereA=A=A+∪-A+whereA+=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 P_{5} 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 l_{1} = 1 and l_{2} = 3.
New graph designs
H
B
λ
F
H
B
λ
F
K_{6t, 6t}
{C6}
2
K_{2}
K_{5t, 5t}
{P6,P4∪2K2}
3
K_{2}
K_{10t, 10t}
{C10}
2
K_{2}
K_{6t, 6t}
{C6,P5∪2P2}
3
K_{2}
K_{8t, 8t}
{C4}
3
K_{2}
K_{6t, 6t}
{C6,G}
3
K_{2}
K_{6t, 6t}
{P7}
3
K_{2}
K_{8t, 8t}
{C6}
3
K_{2}
K_{6t, 6t}
{C4∪S2}
3
K_{2}
K_{n, n}
{P4}
3
K_{2}
K_{6t, 6t}
{C6,P4∪P3∪P2}
3
K_{2}
K_{5t, 5t}
{C4∪K2,G}
3
K_{2}
K_{6t, 6t}
{C6,P5∪2P2}
3
K_{2}
K_{5t, 5t}
{P6}
3
K_{2}
K_{8t, 8t}
{C6}
3
K_{2}
K_{5t, 5t}
{P6,P4∪2K2}
3
K_{2}
K_{n, n}
{P4}
3
K_{2}
K_{5t, 5t}
{P4∪P3,P5∪K2}
3
K_{2}
K_{5t, 5t}
{P6}
3
K_{2}
K_{5t, 5t}
{P4∪K2}
3
K_{2}
K_{5t, 5t}
{P6,P4∪2K2}
3
K_{2}
H
{P4}
3
K_{2}
H
{P5}
4
K_{2}
H
{P5}
3
K_{2}
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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