Ratio Mathematica Volume 42, 2022 A study on the number of edges of some families of graphs and generalized Mersenne numbers Sreekumar K G∗ Ramesh kumar P† Manilal K ‡ Abstract The relationship between the Nandu sequence of the SM family of graphs and the generalized Mersenne numbers is demonstrated in this study. The sequences obtained from the peculiar number of edges of SM family of graphs are known as Nandu sequences. Nandu se- quences are related to the two families of SM sum graphs and SM bal- ancing graphs. The SM sum graphs are established from the inherent relationship between powers of 2 and natural numbers, whereas the SM balancing graphs are linked to the balanced ternary number sys- tem. In addition, some unusual prime numbers are discovered in this paper. These prime numbers best suit as an alternate for the Mersenne primes in the case of the public key cryptosystem. Keywords: nthSM balancing graphs, nthSM sum graphs, Nandu Sequence, generalized Mersenne numbers, bipartite Kneser type-1 graphs. 2020 AMS subject classifications: 05C99, 40A30.1 ∗Department of Mathematics, University of Kerala, Thiruvananthapuram, India; sreeku- mar3121@gmail.com. †Department of Mathematics, University of Kerala, Thiruvananthapuram, India; ramesh.ker64@gmail.com. ‡Department of Mathematics, University College, University of Kerala, Thiruvananthapuram, India.; manilalvarkala@gmail.com. 1Received on January 12th, 2022. Accepted on May 12th, 2022. Published on June 30th, 2022. doi: 10.23755/rm.v39i0.704. ISSN: 1592-7415. eISSN: 2282-8214. c©The Authors. This paper is published under the CC-BY licence agreement. 61 Sreekumar K G, Ramesh Kumar P, and Manilal K. 1 Introduction In computer science, number systems and related ideas are used, particularly in cryptography. The binary number system, which is used in binary computers, and the balanced ternary number system, which is used in ternary computers, are two number systems used in computers. The balanced ternary number system was used in the Russian SETUN computer. Graph theory is used to investigate the combinatorial structure of these two number systems. SM sum graphs and SM balancing graphs are the two categories of graphs based on these number sys- tems that we will focus on here. These graphs are groups of graphs that have been structured in a specific order. SM family of graphs consists of SM sum graph, SM balancing graphs and its complement graphs. For large values of n, these graphs are all non-asymmetric graphs [10] with bigger automorphism groups. The prop- erties of these graphs lead to the discovery of some classical sequences. These sequences are called Nandu sequences or Ne-sequences. These Nandu sequences have a relation with Mersenne primes as well as generalized Mersenne numbers [8]. Furthermore, this is related to the low weight polynomial form of integers [4] which was used in elliptical curve cryptography. The Residue Number System (RNS) [2, 3] has an important role in modular multiplications in computer sci- ence. This RNS modular multiplication is used in the prime field based in elliptic curve cryptosystems too. Generalized Mersenne numbers are used in RNS mod- ular multiplication for more efficiency. The newly defined sequence {ℵn} in this paper are the particular cases of the generalized Mersenne numbers. At present, the modular multiplication requires a maximum of modulus 521. Eventually, the relationship between the Nandu sequence and the generalized Mersenne numbers are established in this work. The use of this relationship in elliptic curve cryptog- raphy is yet to be worked out. Also, a study on Mersenne primes in real quadratic fields was done by Sushma Palimar and B.R. Shankar [6]. The primality testing of large numbers in Maple was given in the work of S.Y.Yan [12]. A study on low weight polynomial form of integers for efficient modular multiplication was done by Jaewook Chung and M.Anwar Hasan [4]. A study on generalized double Fibonomial numbers was done by Mansi shah and Devbhadra Shah [7]. In this paper, we established some properties of Nandu sequences. The Nandu se- quence {Nt∑ n } for SM( ∑ n) satisfies the recurrence relation Nt ∑ n+1 = 2Nt∑ n + 2n + n− 1, Nt∑ 2 = 2. Let ℵn = NtBn −Nt∑n n , where NtBn is the Nandu se- quence of SM(Bn). We derived a closed form of the generating function of the sequence ℵn and is given by G(x) = n−2∑ r=0 2n−2 ( 3 2 )r xr. The convergence of ∑ 1 ℵn is then obtained. Some preliminaries are given below. 62 A study on number of edges and generalized Mersenne numbers 2 Preliminary In this section, we provide the basic definitions and some results from the related work mainly from [9, 10]. We begin with the definition of SM families of graphs. Let’s look at how SM balancing graphs are defined [9]. Consider the set Tn = {3m : m is an integer, 0 ≤ m ≤ n− 1} for a fixed positive integer n ≥ 2. Let I = {−1, 0, 1}. Let x ≤ 1 2 (3n − 1) be any positive integer which is not a power of 3. Then x can be expressed as x = n∑ j=1 αjyj, (1) where αj ∈ I, yj ∈ Tn and y′js are distinct. Each yj such that αj 6= 0 is called a balancing component of x. Consider the simple digraph G = (V,E), where V = {v1,v2, . . . ,v1 2 (3n−1)} and adjacency of vertices is defined by: for any two distinct vertices vx and vyj , (vx,vyj ) ∈ E if (1) holds and αj = −1, and (vyj,vx) ∈ E if (1) holds and αj = 1. This digraph G is called the nthSMD balancing graph, denoted by SMD(Bn). Its underlying undirected graph is called the nth SM balancing graph, denoted by SM(Bn). Let us now look at the definition of SM sum graphs. If p < 2n, is a positive integer which is not a power of 2, then p = ∑n 1 xi, with xi = 0 or 2 m, for some integer m, 0 ≤ m ≤ n− 1 and xi’s are distinct. Here each xi 6= 0 is called an additive component of p. For a fixed integer n ≥ 2, the simple graph SM( ∑ n), called n th SM sum graph [9], is a graph with vertex set {v1,v2, . . . ,v2n−1} and adjacency of vertices defined by, vi and vj are adjacent if either i is an additive component of j or j is an additive component of i. In SM( ∑ n), degree of the vertex v2n−1 is n and ∑ v∈V deg v = 2n(2n−1 − 1). In SM(Bn), the number of vertices is 1 2 (3n − 1) and ∑ v∈V deg v = 2n(3n−1 − 1). Note: For a fixed integer n ≥ 2, let Tn = {3m : m is an integer, 0 ≤ m ≤ n − 1}, N = {1, 2, 3, . . . , t}, where t = 1 2 (3n − 1). Also, let Pn = {2m : m is an integer, 0 ≤ m ≤ n − 1}, M = {1, 2, 3, . . . , 2n − 1}. Then consider Pcn = M −Pn , Tcn = N −Tn throughout this paper unless otherwise specified. The Hamming weight of a string was defined as the number of 1’s in the strings of 0 and 1. Here the number of additive components gives the Hamming weight of string (binary) representation of all numbers in Pcn. The Hamming weight of string (binary) representation of numbers in Pn is always 1. 63 Sreekumar K G, Ramesh Kumar P, and Manilal K. Bipartite Kneser type-1 graphs Let Sn = {1, 2, · · · ,n}, for an integer n > 1. For any two integers k ≥ 1 and n ≥ 2k + 1, the bipartite Kneser graph [10] H(n,k) has all the k-element subsets and all the (n − k)-element subsets of Sn as vertices, and two vertices are adjacent if and only if one of them is a subset of the other. Here we define the bipartite Kneser type-1 graph as follows. Definition 2.1. [10] Let Sn = {1, 2, 3, . . . ,n} for a fixed integer n > 1. Let φ(Sn) be the set of all non-empty subsets of Sn. Let V1 be the set of 1- element subsets of Sn and V2 = φ(Sn) −V1. Define a bipartite graph with adjacency of vertices as: a vertex A ∈ V1 is adjacent to a vertex B ∈ V2 if and only if A ⊂ B. This graph is called a bipartite Kneser type-1 graph. This bipartite Kneser type-1 graph is isomorphic to the graph SM( ∑ n) for each n. To study the structure of the bipartite Kneser type-1 graph, we can make use of SM( ∑ n) graph. The automorphism groups of the bipartite Kneser type-1 graphs are isomorphic to the symmetric group Sn for each n > 2. 3 Nandu sequences of SM( ∑ n) and SM(Bn) The Nandu sequence or Ne-sequence {Ntm}n−1m=1 of SM graphs are the se- quence of numbers whose terms are the half of the sum of degrees of the vertices of SM( ∑ n) or SM(Bn) for all n ≥ 2. Here we assume n ≥ 2 for both the SM sum graph and SM balancing graphs unless otherwise specified. Kinkar Das and I Gutman [5] estimated the Wiener index by means of number of vertices, number of edges, and diameter. 3.1 Nandu sequence for the graph SM( ∑ n) Definition 3.1. For the SM sum graph SM( ∑ n), with vertex set V = {vi : 1 ≤ i ≤ 2n − 1}, the Nandu sequence {Nt∑ n } is defined as a sequence with nth term as Nt∑ n = 1 2 ∑ v∈V deg v and the sequence {DNt∑ n } defined by DNt∑ n =∑ v∈V deg v as double Nandu sequence. i.e., {Nt∑ n } = 2, 9, 28, 75, 186, . . . . Theorem 3.1. Let {Nt∑ n } be the Nandu sequence for the SM sum graph SM( ∑ n), n ≥ 2. Then Nt∑ n+1 = 2Nt∑ n + 2n + n− 1, Nt∑ 2 = 2. 64 A study on number of edges and generalized Mersenne numbers Proof. Consider the graph SM( ∑ n) with vertex set V = {vi : 1 ≤ i ≤ 2 n − 1}. The Nandu sequence is {Nt∑ n } with Nt∑ n = 1 2 ∑ v∈V deg v. Then we have Nt∑ n = n(2n−1 − 1). Nt∑ n+1 = (n + 1)(2n − 1) = n(2n − 1) + 2n − 1 = 2n(2n−1 − 1) + 2n − 1 + n = 2Nt∑ n + 2n + n− 1. Hence proved. Theorem 3.2. Let Nt∑ n be a Nandu sequence of SM Sum graph. Then the fol- lowing holds. 1. Nt∑ n is a composite number for all n > 2 and is always divisible by n. 2. If Nt∑ n n is a prime, then n− 1 is a prime. 3. 1 2n ∑ v∈V deg v = Nt∑ n n is a Mersenne number. 4. Nt∑ n n gives the number of times each element of Pn is used to make num- bers in Pcn, the complement of Pn for a fixed n. Proof. The proof is obvious from the definition of the sequence Nt∑ n . Definition 3.2. Let V be the vertex set of G = SM( ∑ n). Let ∆ be the maximum degree of G and δ be the minimum degree of G. The vertex degree polynomial of G is given by Deg(G,x) = ∆∑ m=δ deg(G,m) ·xm = n∑ k=2 ( n k ) xk + n ·x2n−1−1, where deg(G,m) is the number of vertices of degree m. Let G = SM( ∑ n) be an SM sum graph with vertex set V . The derivative of Deg(G,x) at x = 1 is DNt∑ n , the (n− 1)th term of the double Ne-sequence of G. Now let us see the summation of terms of Nandu Sequence of SM sum graph. Let {Nt∑ n }, where Nt∑ n = 1 2 ∑ v∈V deg v, be the Nandu sequence for the SM sum graphs. Then its summation is given by n∑ r=1 Nt∑ r = n2n+1 − n(n+1) 2 −n. 65 Sreekumar K G, Ramesh Kumar P, and Manilal K. Lemma 3.1. [9] If G = SM( ∑ n), Pn = {2 m : m is an integer, 0 ≤ m ≤ n−1}, then d(vi, vj) =   1 if i is an additive component of j or j is an additive component of i, 2 if i, j ∈ Pn or i, j 6∈ Pn, i and j have at least one common additive component, 3 if neither i nor j is an additive component but exactly one of them belongs to Pn, 4 if i, j 6∈ Pn, i and j have no common additive components. Proposition 3.1. [9] Let G = SM( ∑ n) be an n th SM sum graph. Let dr(vi,vj) denote the number of unordered pairs of vertices for which d(vi,vj) = r. Then dr(vi,vj) =   n.(2n−1 − 1) if r = 1, n(n− 1) 2 + [ (2n −n− 2)(2n −n− 1) 2 − δn ] if r = 2, (n + 1) · 2n − (n + 2)2n−1 −n2 if r = 3, δn if r = 4, where δn = 1 2 n−2∑ r=2 [( n r )n−2∑ k=2 ( n−r k )] . Remark 3.1. For n = 2 or 3, we get that δn = 0. In these cases, the diameter of the graph SM( ∑ n) is 2 or 3. Theorem 3.3. Suppose G = SM( ∑ n) n ≥ 2 be an n th SM sum graph. The (n− 1)th term of the Nandu sequence is equal to d1(vi,vj), where vivj is an edge of G. Proof. The proof follows from Proposition 3.1 . 3.2 Nandu sequence for the graph SM(Bn) We introduced two new sequences, called Nandu sequence and Double Nandu sequence for the SM balancing graphs also. Here we discuss some of their prop- erties. Proposition 3.2 ([9]). For the nth SM Balancing graph SM(Bn), let dr(vi,vj) be the number of unordered pairs of vertices for which d(vi,vj) = r. Let t = 1 2 (3n − 1). Then dr(vi,vj) =   n · (3n−1 − 1) if r = 1, n(n− 1) 2 + [ (t−n)(t−n− 1) 2 −σn ] if r = 2, 1 2 (n · 3n−1 + n− 2n2) if r = 3, σn if r = 4, 66 A study on number of edges and generalized Mersenne numbers where σn = 1 2 n−2∑ r=2 [( n r )n−2∑ k=2 ( n−r k ) 2r+k−2 ] . Definition 3.3. Consider the SM balancing graph SM(Bn), n ≥ 2 , with ver- tex set V = {vi : 1 ≤ i ≤ 12 (3 n − 1)}. The sequence {NtBn} with NtBn = 1 2 ∑ v∈V deg v is called the Nandu sequence and the sequence {DNtBn} defined by DNtBn = ∑ v∈V deg v is called the double Nandu sequence. i.e.,{NtBn} = 4, 24, 104, 400, 1452, . . . . Definition 3.4. Let G = SM(Bn) with vertex set V . Let ∆ be the maximum degree of G and δ be the minimum degree of G. The vertex degree polynomial of G is given as Deg(G,x) = ∆∑ m=δ deg(G,m) ·xm = n∑ k=2 2k−1 ( n k ) xk + n ·x3 n−1−1. Example 3.1. For n = 5, Deg(G,x) = 16x5 + 40x4 + 40x3 + 20x2 + 5x80. Theorem 3.4. Let G = SM(Bn) be a SM balancing graph with vertex set V . The derivative of Deg(G,x) at x = 1 is DNtBn , the (n− 1)th term of the double Ne-sequence of G. Proof. Let Deg(G,x)′ be the derivative of Deg(G,x) w.r.to x. Deg(G,x)′ = n(2x + 1)n−1 −n + n · (3n−1 − 1) ·x3 n−1−2 Hence, Deg(G, 1)′ = n · 3n−1 −n + n · (3n−1 − 1) = 2n(3n−1 − 1) = DNtBn. Here we provide the summation for the Nandu sequence of SM balancing graphs. Let {NtBn}, where NtBn = 1 2 ∑ v∈V deg v, be the Nandu sequence for the SM balancing graphs. Then n∑ r=1 NtBr = 3 4 [ 2n · 3n + 3n − 1 ] −n− n(n+1) 2 . Theorem 3.5. Let SM(Bn) be the nth SM balancing graph and NtBn be the Nandu sequence. Then NtBn+1 = 3NtBn + 3n + 2n−1, NtB2 = 4, for all n ≥ 2. 67 Sreekumar K G, Ramesh Kumar P, and Manilal K. Proof. Consider the graph SM(Bn). Then we have NtBn = n(3 n−1 − 1), NtB2 = 4. For all n ≥ 2, NtBn+1 = (n + 1)(3 n − 1) = n(3n − 1) + 3n − 1 = 3n(3n−1 − 1) + 3n − 1 + n = 3NtBn + 3 n + 2n− 1. Hence proved. 4 Relationship between Nandu sequence and gener- alized Mersenne numbers Mersenne numbers are numbers of the form Mn = 2n−1. If a Mersenne num- ber is prime, then n is a prime. But the converse is not true. Mersenne numbers are a particular case of a larger class of numbers, the generalized Mersenne num- bers [11], Ga,n = an−(a−1)n characterised by their base a and exponent n. The idea of generalized Mersenne numbers was introduced by Solinas [8] in 1999 for the use in elliptic curve cryptography. The use of generalised Mersenne numbers in modular arithmetic to perform fast modular multiplications is well known. Theorem 4.1. Let NtBn and Nt∑n be the terms of the Nandu sequence of SM balancing graph and SM sum graph respectively. Then NtBn−Nt∑n = n(3n−1− 2n−1). Proof. The proof follows from the definition of Nandu sequences of SM sum graph and SM balancing graphs. Definition 4.1. Let SM( ∑ n) and SM(Bn) be the SM sum graphs and SM bal- ancing graphs respectively, n ≥ 3. Then the sequence {ℵn} is defined as ℵn = NtBn −Nt∑n n . Theorem 4.2. For n ≥ 3 and when n is odd, ℵn ≡ 0( mod 5) Proof. We have ℵn = NtBn −Nt∑n n = 3n−1 − 2n−1 Since n is odd, then n− 1 is even, say n− 1 = 2m, for some integer m. Also, 3n−1 − 2n−1 = 32m − 22m, which is a multiple of 5. Hence proved. Theorem 4.3. ∑ 1 ℵn converges. 68 A study on number of edges and generalized Mersenne numbers Proof. Suppose SM( ∑ n) and SM(Bn) be the SM sum graphs and SM balancing graphs respectively. Then we have the sequence {ℵn} as ℵn = NtBn −Nt∑n n . Therefore ℵn = 3n−1 − 2n−1 = (3 − 2)(3n−2 + 3n−3 · 2 + · · · + 2n−2) ≥ 2n−2 + 2n−2 + · · · + 2n−2 = (n− 1) · 2n−2 But 1 3n−1 − 2n−1 ≤ 1 (n− 1).2n−2 = 4 (n− 1) · 2n . Now consider the series ∑ 1 (n− 1) · 2n . We have 2n > n− 1 , for n ≥ 3. To check the convergence of ∑ 1 (n− 1) · 2n , it is enough to check the conver- gence of ∑ 1 (n− 1)2 . But ∑ 1 (n− 1)2 is convergent. Therefore, by comparison test, ∑ 1 ℵn converges. Theorem 4.4. Let SM( ∑ n) and SM(Bn) be the SM sum graphs and SM bal- ancing graphs. Then ℵn+1 = 2ℵn + 3n−1, for all n ≥ 2, given ℵ2 = 1. Proof. Consider the graph SM( ∑ n) and SM(Bn). Then we have ℵn = NtBn −Nt∑n n . ℵn+1 = 3n − 2n = 3.3n−1 − 2.2n−1 = 2(3n−1 − 2n−1) + 3n−1 = 2ℵn + 3n−1. Hence proved. Lemma 4.1. A closed form of the generating function of the sequence ℵn is given by G(x) = n−2∑ r=0 2n−2 ( 3 2 )r xr. Proof. We have ℵn+1 = 2ℵn + 3n−1, for all n ≥ 2, given ℵ2 = 1. When n = 2, ℵ3 = 2ℵ2 + 3 ℵ4 = 2ℵ3 + 32=2(2ℵ2 + 3) + 32=22ℵ2 + 2 · 3 + 32 Similarly, ℵ5 = 2ℵ4 + 32=23ℵ2 + 22 · 3 + 2 · 32 + 33 ℵ6 = 24ℵ2 + 23 · 3 + 22 · 32 + 2 · 33 + 34. 69 Sreekumar K G, Ramesh Kumar P, and Manilal K. Continuing in this way we get, ℵn = n−2∑ r=0 2n−2 ( 3 2 )r Therefore, the required generating function is G(x) = n−2∑ r=0 2n−2 ( 3 2 )r xr. Theorem 4.5. Let SM( ∑ n) and SM(Bn) be the SM sum graphs and SM bal- ancing graphs. Then ℵn+1 = G3,n. Theorem 4.6. If ℵn is prime, then n is even. But the converse is not true. Proof. The proof follows from Theorem 4.2. Definition 4.2. Let SM( ∑ n) and SM(Bn) be the SM sum graphs and SM bal- ancing graphs. Then ℵn+1 = G3,n. For some values of n, G3,n is a prime. These prime numbers are called SM prime numbers. These prime numbers can be used to replace the Mersenne primes in the new public key cryptosystem introduced by D. Aggarwal, et al [1]. In their work, they propose a new public-key cryptosystem whose security is based on the computa- tional intractability of the problem: Given a Mersenne number p = 2n−1, where n is a prime, a positive integer h and an n-bit integer H, decide whether there exist n- bit integers F, G each of Hamming weight less than h such that H = F G modulo p. Theorem 4.7. The series ∑ 1 Nt∑ n and ∑ 1 NtBn converges. Proof. We have for n ≥ 3, 2n − 1 > n. So ∑ 1 n(2n − 1) ≤ 1 n2 . As ∑ 1 n2 is convergent, ∑ 1 n(2n − 1) = ∑ 1 Nt∑ n is convergent. The same way ∑ 1 NtBn also converges. Hence proved. We observed that ℵn is always an odd number and is a prime when n = 3, 4, 6, 18, 30 and 32 so on. There exists G3,n primes. Currently, the largest mod- ulus required for modular multiplication is 521. For n = 6, ℵn = 665. Consider the function AG3,n (x) as a function which gives the number of SM primes among the G3,n generalized Mersenne numbers which are less than or equal to the corresponding ℵn. We get that for n = 2, AG3,n (x) = 1, for n = 3, AG3,n (x) = 2, for n = 4, AG3,n (x) = 3 and for n = 5, AG3,n (x) = 3 etc. In fact the sum of degrees of vertices vx, x ∈ Tcn minus the sum of degrees of vertices vx, x ∈ Pcn is the same as the sum of degrees of vertices vx, x ∈ Tn minus the sum of degrees of vertices vx, x ∈ Pn. If these difference on either side is divided by n, then the quotient is equal to the ℵn. 70 A study on number of edges and generalized Mersenne numbers Theorem 4.8. If p1,p2,p3, ...,pn are some odd prime numbers, then P = ∏n i=1 pi is having 1 as an additive component. Proof. Since p1,p2,p3, ...,pn are odd prime numbers, each is having 1 as an addi- tive component. Then clearly P = ∏n i=1 pi also has 1 as an additive component. This proves the theorem. Corolary 4.1. Let the number of odd prime numbers less than 2n be denoted by N(β). Then N(β) ≤ 2n−1 − 1 for all n ≥ 3. 5 Conclusion The Nandu sequences of the two SM families of graphs were examined, and a relation between these sequences and the generalized Mersenne numbers was established. As a consequence, we have a new sequence of integers called G3,n, which is a type of generalized Mersenne number that may be employed in ellip- tical curve cryptography. These sequences are important in using a graph theory method to investigate the nature and structure of the two number systems - bi- nary and balanced ternary. It has been noted that the (n− 1)th term of the Nandu sequence of SM( ∑ n) is identical to unordered pairs of vertices which are at dis- tance one. The ℵn functions can be studied further in relation to elliptic curve cryptography. Conflict of Interest The authors hereby declare that we have no potential conflict of interest. 6 Acknowledgements The authors would like to thank the anonymous referee for their helpful com- ments which greatly improved the paper. This research has been promoted/supported by the University of Kerala, India. References [1] Divesh Aggarwal, Antoine Joux, Anupam Prakash, and Miklos Santha, A new public-key cryptosystem via mersenne numbers, Annual International Cryptology Conference, Springer, 2018, pp. 459–482. 71 Sreekumar K G, Ramesh Kumar P, and Manilal K. [2] J-C Bajard and Laurent Imbert, A full rns implementation of rsa, IEEE Trans- actions on computers 53 (2004), no. 6, 769–774. [3] Jean-Claude Bajard, Marcelo Kaihara, and Thomas Plantard, Selected rns bases for modular multiplication, 2009 19th IEEE Symposium on Computer Arithmetic, IEEE, 2009, pp. 25–32. 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