Abstract
The representation of the action of on in a graphical format is labeled as coset diagram. These finite graphs are acquired by the contraction of the circuits in infinite coset diagrams. A circuit in a coset diagram is a closed path of edges and triangles. If one vertex of the circuit is fixed by, then this circuit is titled to be a length- circuit, denoted by. In this manuscript, we consider a circuit of length 6 as with vertical axis of symmetry, that is, . Let and be the homomorphic images of acquired by contracting the vertices and , respectively, then it is not necessary that and are different. In this study, we will find the total number of distinct homomorphic images of by contracting its all pairs of vertices with the condition The homomorphic images are obtained in this way having versatile applications in coding theory and cryptography. One can attain maximum nonlinearity factor using this in the encryption process.
1. Introduction
It is prominent that the finite presentation is known as the modular group generated by the linear fractional transformations . In [1], Akbas discussed suborbitalgraphs for the modular group by showing that these graphs contains no circuit if and only if it contains no triangles. If we insert an extra generator with and, another group is emerged, denoted as [2], an extension of with the finite presentation as
In 1978, Professor Graham Higman propounds an unfamiliar type of a graph, titled as coset diagram, which presents the action of on , where is a finite field and shows a prime power. In 1983, this foundation is laid by Qaisar Mushtaq [3]. Small triangles are proposed for the cycle , such that permutes the vertices of triangles in the opposite direction of rotation of clock and an edge is attached to any two vertices that are interchanged by . Heavy dots represent the fixed points of and . Note that equals , which means reverses the triangle orientation proposed for the cycle . For that reason, the diagram need not to be made more perplexing by interjecting edges.
A coset diagram (subdiagram) is said to be a homomorphic image of the coset diagram (subdiagram) if and only if with, where , there exist a vertex in such that .
Coset diagrams obtained from the action of over are infinite graphs [4], where . These diagrams are not easy to study because they are infinite. Thence, coset diagrams are considered as important for the action of on because this action presents finite graphs. The number is an expression of the number , where These finite coset diagrams are the homomorphic images of the coset diagrams for , where for some .
To explain more, coset diagram in Figure 1 illustrates the action on by with permutation representations , and by , , and, respectively, as
Thus, gives that the coset diagram is a homomorphic image of the coset diagram for .
Coset diagrams obtained from the action of on have some attractive narrative. In [5], the quadratic irrational numbers are classified by taking prime modulus that proved helpful in investigating the modular group action on the real quadratic field. The number is called the conjugate of , where and are integers and is a fixed number from , which is not a perfect square. is said to be an ambiguous number [6], if the sign of is different from the sign of . is said to be a totally negative (positive) if and both have the same signs. For a fixed , the number of ambiguous numbers of the form is finite and that segment of the coset diagram attained by the ambiguous numbers forms a closed path (circuit) and it is the only closed path in -orbit [4]. With the help of coset diagram, Anna Torstensson not only described the applications to study the finitely presented group but also discussed the one-relator quotients of the modular group [7].
A closed path of triangles and edges in a coset diagram is called a circuit. In a coset diagram, a circuit is said to be a length- circuit, denoted by , if its one vertex is fixed by
Alternatively, it means that one vertex of the triangles lies outside of the circuit and one vertex of the triangles lies inside of the circuit and likewise. Since is a cycle, so it does not matter if one vertex of the triangles lies inside of the circuit and one vertex of the triangles lies outside of the circuit and likewise. Note that is always even.
The circuit of the type is termed as a periodic circuit with period of length .
Note 1. By we mean the collection of vertices lies on the circuit .
Let be any two vertices fixed by the words and , that is, and . Suppose is the word that maps to , then Note that and are the only two paths that assign to . Now, by contraction of the pair of vertices and , we mean that and melt together to become one node such that . As a result of this contraction, a closed path is created that contains the vertex fixed by and . This closed path is the homomorphic image of the circuit . It is important to note that and is not the only pair of contraction in that creates homomorphic image . There are also many pairs of contraction other than and that create the same homomorphic image . The following theorems proved in [8] will help us to find the total number of such contracted pairs that produce the same homomorphic image of .
Theorem 1. Let the vertices and in are contracted and a homomorphic image of is evolved, then is also obtainable if the vertices and in are contracted.
Theorem 2. If and are contracted to obtain , then during this process number of pairs are contracted all together, where is the collection of words such that and are contained by .
Example 1. Let us contract the vertices and from the circuit (Figure 2) and acquire a homomorphic (Figure 3) of the circuit . Thus, and are the two possible paths between and that are fixing the vertex in .
Let be the family of words such that for all implies and lie on the circuit , then . Then by Theorem 2, the cardinality of implies that there are 9 pairs of vertices contracted to generate the homomorphic image .
Note that the cardinality of does not give the total number of contracted pairs to generate the homomorphic image . In the following, we will discuss the process to find the total number of contracted pairs to generate .
Let denote itself as the mirror image of . Thus, the permutation ensures that the coset diagram is symmetric along the vertical axis. This implies will assuredly occur.
If is a word, then If the word fixes the vertex , then the vertex is fixed by .
A homomorphic image has a symmetry with respect to vertical axis if and only if by contracting and , the vertices and are also contracted.
Remark 1. In coset diagrams, reverses the orientation of the triangles representing the three cycles of (as reflection does). So corresponding to each vertex fixed by the pair , , there is a vertex in (mirror image of ) such that is a fixed point of , . In other words, it is created by contracting and . There are certain ’s which have a vertical symmetry and so have the same orientations as those of their mirror images. The homomorphic image of a circuit , which has a vertex fixed by the pair , , has the same orientation as that of its mirror image if and only if there is a vertex in such that .
1.1. Counting the Number of Pairs of Contracting Vertices of a Homomorphic Image
Let be a homomorphic image of the circuit acquired by the contraction of pair of vertices and of . Then by Theorem 2, has number of pairs of vertices. To find the total number of pairs of vertices, one should follow the following steps.
To know how many total pairs of contracting vertices are there, special precaution must be taken.(1)If by contracting and to create , the pair of vertices and are not contracted, then has different orientation from its mirror image So, there are number of more pairs of vertices for the mirror image of .(2)If by contracting and to create , the pair of vertices and are also contracted, then has the same orientation as that of its mirror image So, in this case, has number of pairs of contracted vertices.
Consider a circuit of length 6 as (Figure 4) with vertical axis of symmetry, that is, . Suppose . The coset diagrams are composed of circuits. The vertices of the circuits in infinite diagrams are contracted in a certain way, and a finite coset diagram evolves. It is therefore necessary to ask how many distinct homomorphic images are obtained if we contract all the pairs of vertices of the circuit ? We not only give the answer to this question for a circuit but also mention those pairs of vertices which are “important”. There is no need to contract the pairs which are not mentioned as “important”. If we contract those, we obtain a homomorphic image, which we have already obtained by contracting “important” pairs.
Note 2. It is clear from Figure 4 that(1)The mirror image of the vertex is , that is, (2)The vertex is fixed by the word (3)The vertex is fixed by the word , where and
Lemma 1. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Proof. Let (Figure 5) be the collection of homomorphic images of acquired by the contraction of the vertices with the vertex in , where and are fixed by the words and . It is easy to verify that and are the possible paths between and (Figure 4). This implies that for each , the homomorphic image has a vertex fixed by and .
Thus,is the family of elements in such that contains the vertices and . This gives that the cardinality of , that is, is the number of contracted pairs of vertices to produce the homomorphic image (Theorem 2). For , let and be any two elements of , then the number of triangles in and are not equal (Figure 5). This implies that all the elements in are different and no one is the mirror image of the other. This further forms the result as . Thence, the number of contracted pairs of vertices of to create all the elements of is .
From Figure 5, it is also clear that no element of except has vertical axis of symmetry. So, is the only homomorphic image whose orientation is not different from its mirror image and all the remaining elements of have different orientations from their mirror images. Hence, there arepairs of contracted vertices to produce all the homomorphic images in .
Lemma 2. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices with the vertex in . Suppose shows itself as the remainder of . Then, graphically we make four partitions of as follows:(i) (Figure 6(a))(ii) (Figure 6(b))(iii) and (Figure 6(c))(iv) and (Figure 6(d))
From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
Let .
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Lemma 3. If we contract the vertices , with the vertex in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices that create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
For a fix value of , let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figures 7–9 present graphically. From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself.
This lemma can be proved by using the same procedure as that for Lemma 1.
Lemma 4. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices with the vertex in . Figure 10 presents graphically. From all the homomorphic images presented in Figure 10, it is not intricated to check that every one is the mirror image of itself.
This lemma can be proved by using the same procedure as that for Lemma 1.
Lemma 5. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Suppose shows itself as the remainder of , then graphically, we make four partitions of as follows:(i) (Figure 11(a))(ii) (Figure 11(b))(iii) and (Figure 11(c))(iv) and (Figure 11(d))
From all the homomorphic images presented in these figures, it is not intricated to check that every one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
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Lemma 6. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Graphically, we make two partitions of as follows:(i) (Figure 12(a))(ii) (Figure 12(b))From all the homomorphic images presented in Figures 12(a) and 12(b), it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
Let.
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Lemma 7. If we contract the vertices , with the vertex in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
For a fix value of , let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figures 13–15 present graphically. From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
Lemma 8. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figure 16 presents graphically. From all the homomorphic images presented in Figure 16, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
Lemma 9. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Suppose shows itself as the remainder of , then graphically we make four partitions of as follows:(i) (Figure 17(a))(ii) (Figure 17(b))(iii) and (Figure 17(c))(iv) and (Figure 17(d))From all the homomorphic images presented in these figures, it is not intricated to check that every one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
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Lemma 10. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Graphically, we make two partitions of as follows:(i) (Figure 18(a))(ii) (Figure 18(b))From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself except . This lemma can be proved by using the same procedure as that for Lemma 1.
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Lemma 11. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figure 19 presents graphically. From all the homomorphic images presented in Figure 19, it is not intricated to check that every one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
Lemma 12. If we contract the vertices, with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Suppose shows itself as the reminder of , then graphically we make four partitions of as follows:(i) (Figure 20(a))(ii) (Figure 20(b))(iii) and (Figure 20(c))(iv) and (Figure 20(d))From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
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Lemma 13. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figure 21 presents graphically. From all the homomorphic images presented in Figure 21, it is not intricated to check that every one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
Lemma 14. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figure 22 presents graphically. From all the homomorphic images presented in Figure 22, it is not intricated to check that no one is the mirror image of itself except . This lemma can be proved by using the same procedure as that for Lemma 1.
Lemma 15. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figure 23 presents graphically. From all the homomorphic images presented in Figure 23, it is not intricated to check that no one is the mirror image of itself.
This lemma can be proved by using the same procedure as that for Lemma 1.
Lemma 16. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Graphically, we make two partitions of as follows:(i) (Figure 24(a))(ii) (Figure 24(b))From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
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Lemma 17. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figure 25 presents graphically. From all the homomorphic images presented in Figure 25, it is not intricated to check that every one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
Lemma 18. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Proof. Let (Figure 26) be the collection of homomorphic images of acquired by the contraction of the vertices with the vertex in , where and are fixed by the words and . It is easy to verify that and are the possible paths between and (Figure 4). This implies that, for each , the homomorphic image has a vertex fixed by and . Thus,is the family of elements in such that contains the vertices and . This gives that the cardinality of , that is, is the number of contracted pairs of vertices to produce the homomorphic image (Theorem 2).
Now, we will prove that all the homomorphic images in are distinct and no one is the mirror image of itself.
Let and be any two elements of , then is produced by contraction of and and is produced by contraction of and . Suppose and are the same. This concludes that is procurable also by contracting and , implying that is one of the pairs of contracted vertices for . Then there must exist an element such that and . is the only element such that but . Thus, and are not the same, that is, by contracting and to produce , and are not contracted. Now, if and are the same, then there must exist an element such that and . But there does not such an element exist in . Thus, is not the same as the mirror image of , that is, by contracting and to create , and are not contracted. This implies that all the elements in are distinct. Thus, gives .
To check does any element of is the mirror image of itself, we suppose is the same to its mirror image . Then by definition, there must exist an element such that and . But there does not such an element exist in . Thus, is not the same as its mirror image, that is, by contracting and to create , and are not contracted. Hence, there arepairs of contracting vertices to produce all the homomorphic images in .
Let .
Lemma 19. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Proof. Let (Figure 27) be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in , where and are fixed by the words and . It is easy to verify that and are the possible paths between and (Figure 4). This implies that, for each , the homomorphic image has a vertex fixed by and . Thus, is the family of elements in such that contains the vertices and . This gives that the cardinality of , that is, is the number of contracted pairs of vertices to produce the homomorphic image (Theorem 2).
Now, we will prove that (i) for , all the homomorphic images in are distinct and no one is the mirror image of itself, and (ii) for , all the homomorphic images in are distinct and only is the mirror image of itself.
Let and be any two elements of , then is produced by contraction of and and is produced by contraction of and .
Suppose and are the same. This concludes that is procurable also by contracting and , implying that is one of the pairs of contracted vertices for . Then there must exist an element such that and . is the only element such that but . Thus, and are not the same, that is, by contracting and to produce , and are not contracted. Now, if and are the same, then there must exist an element such that and . is the only element such that and . This implies that and are the mirror images of each other if and only if and is the mirror image of itself if and only if , that is, . Now,(i)If , then , , implying that . This states that no homomorphic image in is the mirror image of other. Hence, all the homomorphic images in are distinct. Thus, gives . Also, gives that no homomorphic image in is the mirror image of itself. Hence, there are pairs of contracting vertices to produce all the homomorphic images in .(ii)If , then , , implying that . This states that no homomorphic image in is the mirror image of other. Hence, all the homomorphic images in are distinct. Thus, gives .For , we have gives is the mirror image of itself. Hence, there arepairs of contracting vertices to produce all the homomorphic images in .
Let.
Lemma 20. If we contract the vertices , with the vertex in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
For a fix value of , let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figures 28 and 29 present graphically. From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself.
This lemma can be proved by using the same procedure as that for Lemma 18.
Lemma 21. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figure 30 presents graphically. From all the homomorphic images presented in Figure 30, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 18.
Let .
Lemma 22. If we contract the vertices , with the vertex in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
For a fix value of , let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figures 31 and 32 present graphically. From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 18.
Let .
Lemma 23. If we contract the vertices , with the vertices , in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Proof. For , let (Figure 33) be the collection of homomorphic images of acquired by the contraction of the vertices with the vertex in , where and are fixed by the words and . It is easy to verify that and are the possible paths between and (Figure 4). This implies that the homomorphic image has a vertex fixed by and . Thus,is the family of elements in such that contains the vertices and . This gives that the cardinality of , that is, is the number of contracted pairs of vertices to produce the homomorphic image (Theorem 2).
Now, we will prove that all the homomorphic images in are distinct and no one is the mirror image of itself.
Let and be any two elements of , then is produced by contraction of and and is produced by contraction of and .
Suppose and are the same. This concludes that is procurable also by contracting and , implying that is one of the pairs of contracted vertices for . Then there must exist an element such that and . is the only element such that but . Thus, and are not the same, that is, by contracting and to produce , and are not contracted. Now, if and are the same, then there must exist an element such that and . is the only element such that but . This implies that and are not the mirror images of each other, that is, by contracting and to produce , and are not contracted. Hence, all the homomorphic images in are distinct. Thus, gives .
Suppose is the mirror image of itself, then there must exist an element such that and . is the only element such that but . This implies that is not the same as , that is, by contracting and to produce , and are not contracted.
Hence, there arepairs of contracting vertices to produce all the homomorphic images in .
We can prove this lemma for (Figure 34) in similar way as that for .
Let and .
Lemma 24. If we contract the vertices , with the vertices , in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all these homomorphic images and their mirror homomorphic images of is .
Proof. Let (Figure 35) be the collection of homomorphic images of acquired by the contraction of the vertices with the vertices in , where and are fixed by the words and . It is easy to verify that and are the possible paths between and (Figure 4). This implies that the homomorphic image has a vertex fixed by and . Thus,is the family of elements in such that contains the vertices and . This gives that the cardinality of , that is, is the number of contracted pairs of vertices to produce the homomorphic image (Theorem 2).
Now, we prove that (i) for , all the elements in are distinct and no one is the mirror image of itself, and (ii) for , all the elements in are distinct and the homomorphic image is the mirror image of itself.
Suppose and are the same. This concludes that is procurable also by contracting and , implying that is one of the pairs of contracted vertices for . Then there must exist an element such that and . is the only element such that but . Thus, and are not the same, that is, by contracting and to produce , and are not contracted. Now, if and are the same, then there must exist an element such that and . is the only element such that and . This implies that and are the mirror images of each other if and only if and , that is, by contracting and to produce , and are also contracted and the homomorphic image is the mirror image of itself if and only if and that is and . Now,(1)If then , we get implies . This indicates that no homomorphic image in is the mirror image of other. Hence, all the elements in are distinct. Thus, , implying that Also, implies no homomorphic image in is the mirror image of itself. Hence, there are pairs of contracting vertices to produce all the homomorphic images in .(2)If , then only for , and , , implying that . Hence, all the elements in are distinct. Thus, , implying thatNow,(a).If is odd, then , implying that . So, no homomorphic image in is the mirror image of itself. Hence, there are pairs of contracting vertices to produce all the homomorphic images in .(b)If is even, then , implying that is the mirror image of itself. Hence, there are pairs of contracting vertices to produce all the homomorphic images in .Let .
Lemma 25. If we contract the vertices , with the vertices , in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
For a fix value of , let be the collection of homomorphic images of acquired by the contraction of the vertices with the vertices in . Figures 36, 37, and 38 present graphically. From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself.
This lemma can be proved by using the same procedure as that for Lemma 23.
Lemma 26. If we contract the vertices , with the vertices , in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices with the vertices in . Figure 39 presents graphically. From all the homomorphic images presented in Figure 39, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 23.
Let .
Lemma 27. If we contract the vertices , with the vertices , in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Proof. For , let (Figure 40) be the collection of homomorphic images of acquired by the contraction of the vertices with the vertex in , where and are fixed by the words and . It is easy to verify that and are the possible paths between and (Figure 4). This implies that the homomorphic image has a vertex fixed by and . Thus,is the family of elements in such that contains the vertices and . This gives that the cardinality of , that is, is the number of contracted pairs of vertices to produce each homomorphic image (Theorem 2).
Suppose and are the same. This concludes that is procurable also by contracting and , implying that is one of the pairs of contracted vertices for . Then there must exist an element such that and . is the only element such that but . Thus, and are not the same, that is, by contracting and to produce , and are not contracted. Now, if and are the same, then there must exist an element such that and or and . is the only element such that and . This implies that and are the mirror images of each other if and only if and , that is, by contracting and to produce , and are also contracted. For a fix and for all , , implying that and are not the mirror images of each other. This implies that all the homomorphic images in are distinct. Thus, impliesLet be the mirror image of itself, then from Figure 40, we have and , implying . Now, , , implying that is the mirror image of itself. So, out of homomorphic images in , are the mirror images of itself. Hence, there arepairs of contracting vertices to produce all the homomorphic images in . We can prove this lemma for (Figure 41) in similar way as that for .
Let and .
Lemma 28. If we contract the vertices , with the vertices , in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Proof. For , let (Figure 42) be the collection of homomorphic images of acquired by the contraction of the vertices with the vertices in , where and are fixed by the words and . It is easy to verify that and are the possible paths between and (Figure 4). This implies that the homomorphic image has a vertex fixed by and . Thus,is the family of elements in such that contains the vertices and . This gives that the cardinality of , that is, is the number of contracted pairs of vertices to produce each homomorphic image (Theorem 2).
Suppose and are the same. This concludes that is procurable also by contracting and , implying that is one of the pairs of contracted vertices for . Then there must exist an element such that and . are the only element such that and but and . and are not the same, that is, by contracting and to produce , and are not contracted. Now, if and are same, then there must exist an element such that and . is the only element such that and . This implies that and are the mirror images of each other if and only if and , that is, by contracting and to produce , and are also contracted. For all , , implying that and that and are not the mirror images of each other. This implies that all the homomorphic images in are distinct. Thus,impliesLet be the mirror image of itself, then from Figure 42, we have , implying . Now, , , implying that is the mirror image of itself. So, out of homomorphic images in , are the mirror images of itself. Hence, there arepairs of contracting vertices to produce all the homomorphic images in . We can prove this lemma for (Figure 43) in similar way as that for .
Lemma 29. If we contract the vertices , with the vertices , in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
For a fix value of , let be the collection of homomorphic images of acquired by the contraction of the vertices with the vertices in . Figures 44 and 45 present graphically. From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 23.
Let and .
Lemma 30. If we contract the vertices , with the vertices in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
For a fix value of , let be the collection of homomorphic images of acquired by the contraction of the vertices with the vertices in . Figures 46 and 47 present graphically. From all the homomorphic images presented in these figures, it is not intricated to check that if , then is the only homomorphic image which is the mirror image of itself; otherwise, no one is the mirror image of itself.
This lemma can be proved by using the same procedure as that for Lemma 28.
Lemma 31. If we contract the vertices , with the vertices in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
For a fix value of , let be the collection of homomorphic images of acquired by the contraction of the vertices with the vertices in . Figures 48 and 49 present graphically. From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using same procedure as that for Lemma 23.
Let .
Lemma 32. For each , there are 6 pairs of contracting vertices to produce the homomorphic image of by contracting the vertex with the vertex .
For , let be the homomorphic image of acquired by contracting the vertex with the vertex in , where and are fixed by the words and . It is easy to verify that and are the possible paths between and (Figure 4). This implies that the homomorphic image has a vertex fixed by and . Thus,is the family of elements in such that contains the vertices and . This gives that the cardinality of , that is, is the number of contracted pairs of vertices to produce the homomorphic image (Theorem 2). From Figure 50, it is not intricated to check that and its mirror image are not the same. Hence, there arepairs of contracting vertices to produce the homomorphic images .
We can prove this lemma for (Figure 51) in similar way as that for .
Lemma 33. For each , there are 6 pairs of contracting vertices to produce the homomorphic image of by contracting the vertex with the vertex .
For a fix value of , let be the homomorphic image of acquired by the contraction of the vertex with the vertex in . Figures 52 and 53 present graphically. From these figures, it is not intricated to check that is not the mirror image of itself.
This lemma can be proved by using the same procedure as that for Lemma 32.
Let and .
Lemma 34. For each , there are 6 pairs of contracting vertices to produce the homomorphic image of by contracting the vertex with the vertex .
For a fix value of , let be the homomorphic image of acquired by the contraction of the vertex with the vertex in . Figures 54, 55, and 56 present graphically. From these figures, it is not intricated to check that is not the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 32.
Lemma 35. For each , there are 3 pairs of contracting vertices to produce the homomorphic image of by contracting the vertex with the vertex .
For a fix value of , let be the homomorphic image of acquired by the contraction of the vertex with the vertex in . Figures 57(a), 57(b), and 57(c) present and Figure 58 presents graphically. From these figures, it is not intricated to check that is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 32.
Let , thenWe are now able to prove our primary outcome. Letwhere and stand for even and odd, respectively.
(a)
(b)
(c)
Theorem 5. There are number of distinct homomorphic images acquired by the contraction of all pairs of vertices of the circuit .
Proof. Let us collect all the pairs of contracting vertices of mentioned in Lemma 1 to Lemma 35 in the form of set asLet H be the set of homomorphic images obtained by contracting each element of S, thenLet S′ be the sum of contracted pairs obtained by contracting each element of S, thenshows that each element of is contracted.
Now,The value of gives the total number of homomorphic images produced in the contraction of all pairs of vertices of . The value of guaranteed that the set contains all homomorphic images.
2. Conclusion
Thus, there are total number of elements in . To find all distinct homomorphic images, we do not need to contract each pair of vertices of . We have to contract only those pairs of vertices, which are in set and they are in numbers because if we contract the pair which is not belong to set , we attain the homomorphic image, which we already acquired by contracting the element of set .
3. Applications of Homomorphic Copies in Lightweight Cryptography and Chemistry
The construction of any product by using minimum resources without compromising on quality is primary objective of scientists.
The coset diagrams with 256 vertices are used in the construction of strong 8 × 8 S-boxes [9]. In present-day block ciphers, cryptographically secure S-boxes are designed to attain the requirements of Shannon's necessity for perplexity. Substitution boxes are the fundamental segments in numerous Feistel network-based block cryptosystems or substitution-permutation (S-P) networks. The use of 8 × 8 S-boxes in block cipher is excessively costly. So, it is not surprising that we are seeing strong progress in the field of lightweight cryptography in recent years, for instance PRESENT. Lightweight cryptographic calculations are utilized in business items, including DESL, PRINTCIPHER, SEA, HIGHT, PRESENT, LED, and KATAN/KTANTAN. For instance, Keeloq is a 32-bit block cipher frequently utilized in the automobile business. Digital signature transponder is a 40-bit block cipher, implemented in wireless authentication systems. Since a homomorphic copy of a graph is smaller graph with the same algebraic structure, it can be quite handy to generate small sized S-boxes by using homomorphic copies of the coset diagram having 256 vertices.
Interrelation of certain types of coset graphs and structure of carbon allotropes not only highlights the connection but also improvises applications in many fields.
Several forms of carbon can be found in nature. One of the most important allotropes of carbon is fullerene which was discovered in . Fullerenes are carbon-cage-like polyhedral molecules in which a large number of carbon atoms are bonded in a nearly spherical symmetric configuration. Fullerenes Cn can be drawn for and for all even . They have carbon atoms, bonds, 12 pentagonal faces, and hexagonal faces. The most important member of the family of fullerenes is C60. Fullerenes can be classified in terms of their groups of symmetries. These groups are also known as point groups. Every element of a point group is an isometry of the Euclidian space and so it is either a rotation around an axis or it is a reflection in a plane. The list of all 28 fullerene point groups is: Ih, I, Th, Td, T, D6h, D6d, D6, D5h, D5d, D5, D3h, D3d, D3, D2h, D2d, D2, S6, S4, C3h, C2h, C3v, C3, C2v, C2, Cs, Ci, and C1 [10]. Fullerene C60 has icosahedral symmetry, that is, the symmetry group is isomorphic to A5. A constructive enumeration of fullerenes has been dealt in detail in [11]. The structure of coset graphs of fullerene C60, symmetry group and adjacency matrix has been explored through the action of modular group in [12]. Another carbon allotrope with high permutational symmetry is allotrope D168 Schwarzite, proposed by Vanderbilt and Tersoff, which has an automorphism group of order 168. In [13], it has been shown that coset diagram for points has interesting relation with carbon allotrope with negative curvature D168 Schwarzite. The future studies may extend the present study by investigating the homomorphic copies of the coset graphs for and and their related chemical structures.
Data Availability
No datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.