Distance Matrices and Adjacency Matrices of Some Families of Graphs

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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 9 (2019) pp. 2121-2125 c Research India Publications. http://www.ripublication.com

Distance matrices and Adjacency Matrices of Some families of Graphs

Sreekumar.K.G , Manilal. K Department of Mathematics, University College, Thiruvananthapuram, Kerala, India. [email protected] (Published on 14th May 2019)

Abstract Hamming weight of a string was defined as the number of 1s in This paper describes how the distance matrices of some the strings of 0 and 1. Here the number of additive components families of graphs are obtained from their distance formula and is the Hamming weight of string (binary) representation of all c 1 n then the corresponding adjacency matrices. The spectrum and integers in P . Also any positive integer p ≤ 2 (3 −1), which energy of these classes of graphs are based on the adjacency is not a power of 3 can be expressed as a linear combination matrices. We considered mainly two families of graphs - of two or more distinct elements of the set T = {3m : SM sum graphs and SM Balancing graphs. SM sum graphs m is an integer, 0 ≤ m ≤ n − 1} with coefficients −1, 0 or 1. are associated with the intrinsic combinatorial relationship The relationship between p and the elements of T are used to between the powers of 2 and the positive integers, which form a new class of graphs called nthSM Balancing graphs, is used in binary number system. SM balancing graphs are SM(Bn) and SMD(Bn). Some preliminaries are given related with the balanced ternary number system. SM family below. of graphs is vertex labelled graphs. We provide criteria for 1. PRELIMINARY finding the distance matrices of these families of graphs. Some results connecting the sum of entries of these distance matrices Definition 1.1. [6] Consider the set T = {3m : and adjacency matrices are obtained. m is an integer, 0 ≤ m ≤ n − 1} for a fixed integer n ≥ 2. 1 n AMS subject classification: 05C99 Let I = {−1, 0, 1}. Any positive integer x ≤ 2 (3 − 1) which Keywords: nthSM Balancing graphs, nthSM Sum graphs, is not a power of 3 can be expressed as Distance matrix ,Adjacency matrix, Wiener index. Harary n matrix. X x = αjyj (1) j=1 INTRODUCTION for some α ∈ I and y ∈ T . If α 6= 0 , then each y is called The distances in a graph are very important concept in Graph i j i j a balancing component of x. theory. The topological indices of a graph are related to the distances in a graph. The distance between two vertices vi Definition 1.2. [6] Let T be the set T = {3m : and vj, is denoted by d(vi, vj), is defined as the length of the m is an integer, 0 ≤ m ≤ n − 1} for a fixed integer n ≥ 2. shortest path between vi and vj. The distance matrix of a graph Consider the simple directed graph G=(V,E), where the vertex

G having n vertices is a symmetric matrix D = [dij] whose set V = {v1, v2, . . . , v 1 n } and adjacency of vertices 2 (3 −1) entry dij is deﬁned as deﬁned by, for two distinct vertices vx and vyj , vx is adjacent

( to vyj if (1) holds and α = −1 and the vertex vyj is adjacent to d(vi, vj) , if i 6= j th dij = vx if (1) holds and α = 1. This graph G is called the n SMD 0 , if i = j. Balancing Graph, SMD(Bn). The underlying undirected th The two number systems used in computer are the binary graph is called n SM Balancing Graphs, SM(Bn). number systems and the balanced ternary number systems. Definition 1.3. [5] If p < 2n, is a positive integer The fixed length group of binary bits is generally called a which is not a power of 2, then p = Pn x , with computer word. Each k bit computer word can store a number 1 i x = 0 or 2m, for some integer m, 0 ≤ m ≤ n − 1 and as large as 2k − 1, positive or negative. Usually modern i x s are distinct. Here we call each x 6= 0 as an additive processors including embedded systems have a word length i i component of p. of 8, 16, 24, 32 or 64 bits, while modern general purpose computers use 32 or 64 bits. Any positive integer less than Definition 1.4. [5] For a fixed integer n ≥ 2, define a simple 2n and not in P = {2m : m is an integer, 0 ≤ m ≤ n − 1}, P th graph SM( n), called n SM sum graph, with vertex set for a fixed positive integer n ≥ 2, can be expressed as the {v1, v2, . . . , v2n−1} and adjacency of vertices defined by, vi sum of two or more distinct elements of P . If a 6∈ P , a is and vj are adjacent if either i is an additive component of j or n P less than 2 and a = xi, with distinct xi ∈ P , then each j is an additive component of i. xi is called an additive component of a. The simple graph, P n m SM( n) is a graph with vertex set {v1, v2, . . . , v2 −1} and Note: For a fixed integer n ≥ 2, let T = {3 : adjacency of vertices defined by two distinct vertices vx and m is an integer, 0 ≤ m ≤ n − 1}, N = {1, 2, 3, . . . , t} , where vy are adjacent if either x is an additive component of y or t = 1 (3n − 1). Also let P = {2m : m is an integer, 0 ≤ 0 0 2 y is an additive component of x. Also this expression of a m ≤ n − 1}, M = {1, 2, 3,..., 2n − 1}. Then consider is related to the low weight polynomial form of integers [1] P c = M − P , T c = N − T throughout this paper unless which was used in the theory of elliptical curve cryptography. otherwise specified. Let M = {1, 2, 3,..., 2n − 1}, then P c = M − P . The

2121 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 9 (2019) pp. 2121-2125 c Research India Publications. http://www.ripublication.com

P 2. DISTANCES IN THE GRAPHS SM( N ) AND SM(BN ) It is very clear from the SM sum graph that any two odd prime numbers are at distance 2. The distance between 1 and any odd number is 1. Some of the distance related results from the previous work is given below.

P m Lemma 2.1. [5] If G = SM( n), P = {2 : m is an integer, 0 ≤ m ≤ n − 1}, n ≥ 2, then

1 , if i is an additive component of j or j is an additive component of i  2 , if i, j ∈ P or i, j 6∈ P , i and j have atleast one common additive component d(vi, vj) = 3 , neither i nor j is an additive component but exactly one of them belongs to P  4 , i, j 6∈ P , i and j have no common additive component.

P th Proposition 2.2. [5] Let G = SM( n) be an n SM sum graph. Let dr(vi, vj) denote the number of unordered pairs of vertices for which d(vi, vj) = r. Then

n.(2n−1 − 1) , if r = 1  n(n − 1) (2n − n − 2)(2n − n − 1)  + − δ , if r = 2 dr(vi, vj) = 2 2 n n−1 2 (n + 1).2 − (n + 2)2 − n , if r = 3  δ , if r = 4.

n−2 n−2 1 P n P n−r where δ = r k . 2 r=2 k=2

Remark 2.3. δ = 0 for n = 2 or 3

n Pn The value of δ is the number of pairs of pair wise disjoint Let x be a positive integer < 2 . Then x = 1 xi, with m subsets of P excluding the empty set and singleton sets. Here xi = 0 or 2 , for some integer m, 0 ≤ m ≤ n − 1 and P the diameter of the graph SM( n) is given as follows. xis are distinct. Each xi 6= 0 and xi ∈ P is an additive component of x. Let x = Pt a x and y = Pr b x be two  1 i i 1 i i 2 if n = 2 positive integers with a = 0 or 1 and b = 0 or 1, for some  i i diam(G) = 3 if n = 3 positive integers t and r. If the terms in the additive component 4 if n ≥ 4 expansion of x are different from that of y, then x and y are called additive distinct integers. Definition 2.4. [7] Let P = {2m : m is an integer, 0 ≤ m ≤ n − 1} for a fixed integer n ≥ 2. Let x Theorem 2.7. Let Ts(x) be a 2S3 transformation function. n be a positive integer n. Then Pn , with Let x and y be two distinct positive integers < 2 and vx, vy ∈ < 2 x = 1 xi P x = 0 or 2m, for some integer m, 0 ≤ m ≤ n − 1 and V (G), where G = SM( n). Then the number of cases in i n 2 x s are distinct. Each x 6= 0 and x ∈ P is an additive n2 − n − n i i i which Ts(x+y) = Ts(x)+Ts(y) is +δ, where component of x. Let N be the set of all natural numbers. We 2 n−2 n−2 0 1 n n−r define a transformation Ts : N → N such that P P δ = r k . 2 r=2 k=2 n X ∗ Ts(x) = x (2) i Pt Pr 1 Proof. Let x = 1 aixi and y = 1 bixi be two positive integers with a = 0 or 1 and b = 0 or 1, for some integers where N 0 = {1, 2, 3, 4, ..., n} and each x∗ is obtained by i i i t > 0 and r > 0. When x and y are additive distinct integers, changing the base 2 of x to base 3. This transformation is i the terms in the expansion of x are different from that of y. called 2S3 transformation. Then we get Ts(x + y) = Ts(x) + Ts(y). From lemma 2.1, 0 1 2 Example 2.5. Let x = 7 = 2 + 2 + 2 . Then Ts(x) = we get that when d(x, y) is either 3 or 4, then x and y are 30 + 31 + 32 = 13. additive distinct integers. Also when x, y ∈ P , then x and y 2 4 2 4 Also when x = 20 = 2 + 2 . Then Ts(x) = 3 + 3 = 90. are additive distinct integers. Therefore the number of cases in n2n − n2 − n m which T (x + y) = T (x) + T (y) is + δ. This Definition 2.6. Let P be the set P = {2 : s s s 2 m is an integer, 0 ≤ m ≤ n − 1} for a fixed integer n ≥ 2. completes the proof.

2122 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 9 (2019) pp. 2121-2125 c Research India Publications. http://www.ripublication.com

m Lemma 2.8. [6] If the graph G = SM(Bn), T = {3 : m is an integer, 0 ≤ m ≤ n − 1} , vi, vj ∈ V (G) , then

1 , if i is a balancing component of j or j is a balancing component of i.  2 , if i, j ∈ T or i, j 6∈ T , i and j have atleast one common balancing component. d(vi, vj) = 3 , neither i nor j is a balancing component but exactly one of them belongs to T.  4 , i, j 6∈ T , i and j have no common balancing component.

th Proposition 2.9. [6] Let the graph G = SM(Bn) be an n SM Balancing graph. Let dr(vi, vj) denote the number of unordered 1 n pairs of vertices for which d(vi, vj) = r. Let t = 2 (3 − 1). Then

n.(3n−1 − 1) , if r = 1  n(n − 1) (t − n)(t − n − 1)  + − σ , if r = 2 dr(vi, vj) = 2 2 1 n−1 2  (n.3 + n − 2n ) , if r = 3  2 σ , if r = 4.

n−2 n−2 1 P n P n−r r+k−2 where σ = r k 2 2 r=2 k=2

Remark 2.10. σ = 0 for n = 2 or 3

P n i 2.1. Distance Matrices of SM( n) and SM(Bn) P P 2n n 2 Therefore dij= 2 − 3.2 − n + n + 2 + 2δ. i=1 j=1 P Consider the graph G = SM( n), for n ≥ 2 with vertex set Hence proved. n V = {vi : 1 ≤ i ≤ 2 − 1}. The distance matrix of the graph n G having 2 − 1 vertices is a symmetric matrix Dn = [dij] of n order p = 2 − 1, whose entry dij is deﬁned as

( Consider the graph G = SM(Bn), for n ≥ 2 with vertex set d(vi, vj) , if i 6= j d = V = {v1, v2, . . . , v 1 n }. The distance matrix of the graph ij 2 (3 −1) 0 , if i = j, 1 G having (3n − 1) vertices is a symmetric matrix D = [d ] 2 n ij where d(vi, vj) is given in Lemma 2.1. whose entry dij is deﬁned as P th Theorem 2.11. Let G = SM( n) be the n SM sum ( graph, n ≥ 2. Let Dn = [dij] be the distance matrix of d(v , v ) , if i 6= j n i d = i j P P 2n n 2 ij G. Then dij= 2 − 3.2 − n + n + 2 + 2δ, where 0 , if i = j. i=1 j=1 n−2 n−2 1 P n P n−r δ = r k . 2 r=2 k=2 where d(vi, vj) is given in Lemma 2.8. The distance matrix and related matrices of a graph are the

Proof. We have, for the distance matrix, Dn = [dij] for a sources of many graph invariants like topological indices graph, the Wiener index of G, etc. So these matrices are used in structure property activity n i P P modelling by studying the spectra and related polynomials W (G) = dij. of these graphs. The reciprocal distance matrices (Harary i=1 j=1 Also W (G) = P d(u, v), where d(u, v) is the distance matrices) of SM family of graphs is obtained from the {u,v}⊆V corresponding distance matrices by replacing all non-zero between u and v. entries by their reciprocals. Therefore the Harary index of the By the Proposition 2.2, SM sum graphs and SM Balancing graphs are obtained. X W (G) = d(u, v) P Theorem 2.12. Let the graph G = SM( n), n ≥ 2. Let {u,v}⊆V 1 D0 = [d0 ], where d0 = , be the Harary Matrix, then n(n − 1) n ij ij d = 4δ + 3n.2n−1 − n2 + 2 ij 2 (2n − n − 2)(2n − n − 1) + − δ + 1.n.(2n−1 − 1) n n 2 2n n n 2 X X 0 n 2 3.2 n.2 δ dij = + n − 1 + − + − . 2n n 2 3 2 2 3 2 = 2 − 3.2 − n + n + 2 + 2δ i=1 j=1

2123 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 9 (2019) pp. 2121-2125 c Research India Publications. http://www.ripublication.com

Proof. From the proposition 2.2, we get Proposition 2.18. Let G = SM(Bn) , n ≥ 2. Then i) R (v )=3n−1 − 1 , if i ∈ T . n n 1 s i X X 0 X ii) P R (v ) = P R (v ). dij = 2 i∈T s i j∈T c s j dG(u, v) i=1 j=1 {u,v}⊆V P Theorem 2.19. For the SM sum graph SM( n), n ≥ 2, n−1 n−1 = 2n.(2 − 1).1 d(vx, vy) = 2 for all x, y > 2 , x 6= y. n(n − 1) (2n − n − 2)(2n − n − 1) + + − δ 2 2 Proof. When x, y > 2n−1 , x 6= y, n ≥ 2, it is clear that x n−1 2 2 δ and y are not in P and they are having one common additive + n.2 − n . + n−1 3 2 component 2 . Therefore by Lemma 2.1, d(vx, vy) = 2 . n2 22n 3.2n n.2n δ Hence the theorem. = + n − 1 + − + − . 3 2 2 3 2 The transmission Tr(v)[8] of a graph was defined to be the sum of the distances from v to all other vertices. 0 0 Theorem 2.13. Let G = SM(Bn), n ≥ 2. Let Dn = [dij], X Tr(v) = d(u, v) (3) 0 1 where dij = , be the Harary Matrix, u∈V dij n i A graph is said to be k-transmission regular if its distance X X n.3n 7n n2 P then d0 = − − matrix has constant row sums equal to k. The graph SM( ) ij 12 12 6 n i=1 j=1 or SM(Bn) are not transmission regular graph. 3 32n 3n n.3n−1 σ P th Observation 2.20. Let G = SM( n) be an n SM sum + + − + − . 0 0 16 16 4 6 4 graph. Then the graph G with V (G ) = V (G) − {v2n−1}, is a 9-transmission regular graph when n = 3. P 2.2. Adjacency Matrices of SM( n) and SM(Bn) 3. CONCLUSION Definition 2.14. [8] The Adjacency matrix of graph G having In many cases it is difficult to find the distance matrices of p vertices is a symmetric matrix An = [aij] , of order family of graphs of bigger dimensions. But here we provide n p = 2 − 1, whose entry aij is defined as a systematic method for finding the distance matrices of some ( families of graphs- SM family of graphs. The calculation of 1 , if vi is adjcent to vj distance matrix is much easy by using the distance formula aij = 0 , otherwise. for SM graphs of large n. Probably a computer algorithm can be framed by using these idea to get the distance matrix. P The Adjacency matrix of the graph SM( n) with vertex set Then the calculation of cospectrality and other measures will n V = {vi : 1 ≤ i ≤ 2 −1} is obtained from the corresponding be easy. Also the distance matrix is related to the Wiener P distance matrix of SM( n) by replacing all entries which indices of the graphs. So far we considered only the simple are greater than 1 by 0. Similarly we can get the adjacency graph of SM family of graphs. The distance matrices of the matrices of SM(Bn). directed graph of SM family of graphs can be found out in this way. Further study of these distance matrices and adjacency Definition 2.15. Let An = [aij] be the adjacency matrices of P n matrices of these graphs may lead to very useful results. Also the graph SM( n). For each i, 1 ≤ i ≤ 2 −1, define Rs(vi) j=2n−1 the characteristic polynomial of Dn need to be found. P as Rs(vi) = aij. j=1 P Conflict of Interest Proposition 2.16. Let G = SM( n) , n ≥ 2. Then n−1 We hereby declare that the authors have no potential conflict i) Rs(vi)=2 − 1 , if i ∈ P . P P of interest. ii) i∈P Rs(vi) = j∈P c Rs(vj).

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