Adjacency Matrix of Product of Graphs

Transcription

1 Kalpa Publications in Computing Volume 2, 2017, Pages ICRISET2017 International Conference on Research and Innovations in Science, Engineering &Technology Selected Papers in Computing Adjacency Matrix of Product of Graphs H S Mehta 1 and U P Acharya 2 1 Department of Mathematics, Sardar Patel University, V V Nagar, Anand,Gujarat India hs mehta@spuvvnedu 2 Applied Sciences and Humanities Department, A D Patel Institute of Technology, Karamsad, Gujarat, India mathsurvashi@aditacin Abstract In graph theory, different types of matrices associated with graph, eg Adjacency matrix, Incidence matrix, Laplacian matrix etc Among all adjacency matrix play an important role in graph theory Many products of two graphs as well as its generalized form had been studied, eg, cartesian product, 2 cartesian product, tensor product, 2 tensor product etc In this paper, we discuss the adjacency matrix of two new product of graphs G H, where = 2, 2 Also, we obtain the spectrum of these products of graphs Keywords:- 2 cartesian product, 2 tensor product, adjacency matrix and spectrum 2010 Mathematics Subject Classification:- 05C50, 05C76 1 Introduction Graph theory, no doubt, is the fast growing area of combinatory Because of its inherent simplicity Graph theory has a wide range of applications in Computer Science, Sociology, bio-informatics, medicines etc There are different type of product of graphs eg, cartesian product, tensor product etc defined in graph theory There are many matrices associated with graphs like adjacency matrix, incidence matrix, Laplacian matrix etc One of them is adjacency matrix It is well- known that these product operation on graphs and product of adjacency matrices are related ([6], [9]) We have generalized well-known two products, cartesian product and tensor product with the help of concept of distance In this direction, we have defined 2 cartesian product G 2 H and 2 tensor product G 2 H of graphs G and H ([1], [2], [3]) Let G = (V(G),E(G)) be a finite and simple graph For a connected graph G, d G (u,u ) is the length of the shortest path between u and u in G A graph is r regular if every vertex of G has degree r For the basic terminology, concepts and results of graph theory, we refer to ([5], [8]) The Kronecker product A B of two matrices A and B has been defined and discussed in [6] R Buyya, R Ranjan, S Dutta Roy, M Raval, M Zaveri, H Patel, A Ganatra, DG Thakore, TA Desai, ZH Shah, NM Patel, ME Shimpi, RB Gandhi, JM Rathod, BC Goradiya, MS Holia and DK Patel (eds), ICRISET2017 (Kalpa Publications in Computing, vol 2), pp

2 Definition 11 Let A = [a ij ] and B = [b ij ] be in M m n (IR) Then the Kronecker product of A and B, denoted by A B is defined as the partitioned matrix [a ij B], a 11 B a 12 B a 1n B a 21 B a 22 B a 2n B A B = [a ij B] = a m1 B a m2 B a mn B Note that it has mn blocks, where ij th block is the block matrix a ij B of order m n Let X = [x 1,x 2,,x n ] T and Y = [y 1,y 2,,y n ] T be column vectors Then by definition of the kronecker product, we have X Y = [ ] T [ ] T x 1 Y x 2 Y x n Y = x1 y 1 x 1 y m x n y 1 x n y m Definition 12 [6] Let A = [a ij ] and B = [b ij ] be in M m n (IR) Then the Kronecker sum of A and B, denoted by A B and defined as A B = (A I m )+(I n B) Definition 13 [9] Let G = (V,E) be a simple graph with the vertex set u 1,u 2,u n } Then the adjacency matrix A(G) = [a ij ] is defined as follows: a ij = 1; if dg (u i,u j ) = 1 0; otherwise Then A(G) is a real, square symmetric matrix Next, we recall the definitions of 2 tensor product and 2 cartesian product of graphs as follows: Definition 14 [2] Let G = (U,E 1 ) and H = (V,E 2 ) be two connected graphs The 2 tensor product G 2 H of G and H is the graph with vertex set U V and two vertices (u,v) and (u,v ) in V(G 2 H) are adjacent in 2 tensor product if d G (u,u ) = 2 and d H (v,v ) = 2 Note that if d G (u,u ) = 1 = d H (v,v ), then it is the usual tensor product G H of G and H Definition 15[3] Let G = (U,E 1 ) andh = (V,E 2 ) be twoconnectedgraphs The 2 cartesian product G 2 H of G and H is the graph with vertex set U V and two vertices (u,v) and (u,v ) in V(G 2 H) are adjacent if one of the following conditions is satisfied: (i) d G (u,u ) = 2 and d H (v,v ) = 0, (ii) d G (u,u ) = 0 and d H (v,v ) = 2 Note that in the above condition (i) and condition (ii), 2 replace by 1, then it is the usual cartesian product G H of G and H It is clear that the definition of A(G) is in terms of adjacent vertices, ie, vertices at distance 1 For the graphs G H, we use 2 distance between vertices So, we require to consider the second stage adjacency matrix 159

3 Definition 16 [4] Let G = (V,E) be a simple graph with the vertex set u 1,u 2,u n } The second stage adjacency matrix A 2 (G) = [a ij ] is defined as follows: a ij = 1; if dg (u i,u j ) = 2 0; otherwise Note that A 2 (G) is a real, square symmetric matrix 2 Adjacency matrix of G H In this section, we discuss adjacency matrix of G 2 H and G 2 H of graphs G and H Adjacency matrix of usual tensor product and cartesian product of graphs can be obtained as follows Proposition 21 [9] Let G and H be simple graphs with n and m vertices respectively Then (i) A(G H) = A(G) A(H) (ii) A(G H) = (A(G) I m )+(I n A(H)) Next, we prove the result similar to Proposition 21 for G 2 H and G 2 H Proposition 22 Let G and H be connected graphs with n and m vertices respectively Then A(G 2 H) = A 2 (G) A 2 (H) Proof: Let V(G) = u 1,u 2,,u n } and V(H) = v 1,v 2,,v m } A 11 A 12 A 1s A 1n Let A(G 2 H) = A r1 A r2 A rs A rn A n1 A n2 A ns A nn Where A r,s = (u s,v 1) (u s,v 2) (u s,v m) (u r,v 1) P 11 P 12 P 1m (u r,v m) P m1 P m2 P mm Suppose d G (u r,u s ) 2, then A rs is a zero matrix of order m m Suppose d G (u r,u s ) = 2, then P ij = 1; if dh (v i,v j ) = 2 0; otherwise 160

4 Thus, in this case, P ij = (ij) th entry of A 2 (H) So, A rs = A2 (H); if d G (u r,u s ) = 2 0; otherwise Now let A 2 (G) = [b ij ] n n and A 2 (H) = [c ij ] m m Then b 11 A 2 (H) b 1n A 2 (H) A 2 (G) A 2 (H) = b n1 A 2 (H) b nn A 2 (H) So, (ij) th block of A 2 (G) A 2 (H) is the block matrix b ij A 2 (H) of order m m Then the block matrix [b ij A 2 (H)] = A2 (H); if b ij = 1, ie, d G (u i,u j ) = 2 = [A ij ] Therefore, [b ij A 2 (H)] = (ij) th block of A(G 2 H) So, A(G 2 H) = A 2 (G) A 2 (H) Proposition 23 Let G and H be connected graphs with n and m vertices respectively Then A(G 2 H) = [A 2 (G) I m ]+[I n A 2 (H)] Proof: Suppose V(G) = u 1,u 2,,u n } and V(H) = v 1,v 2,,v n } Then consider A(G 2 H) and A rs same as given in Proposition 22 Suppose d G (u r,u s ) = 0 Then P ij = 1; if dh (v i,v j ) = 2 0; if d H (v i,v j ) 2 Suppose d G (u r,u s ) = 2 Then P ij = 1; if i = j 0; if i j Also, suppose d G (u r,u s ) 0 or 2 Then A r,s is a zero matrix of order m m Thus, A 2 (H); if d G (u r,u s ) = 0 A rs = I m ; if d G (u r,u s ) = 2 Next, A 2 (G) = [b ij ] n n and A 2 (H) = [c ij ] m m Then b 11 I m b 12 I m b 1n I m A 2 (G) I m = b n1 I m b n2 I m b nn I m 161

5 So,(ij) th block ofa 2 (G) I m = [b ij I m ] and [b ij I m ] = Im ; if b ij = 1,ie,d G (u i,u j ) = 2 In particular b ii I m = 0 m m A 2 (H) 0 0 I n A 2 (H) = 0 0 A 2 (H) Therefore (ij) th block of I n A 2 (H) is as follows: I n A 2 (H) = A2 (H); if i = j Suppose (ij) th block of [A 2 (G) I m ] [I n A 2 (H)] = B Then A 2 (H); i = j; ie, d G (u i,u j ) = 0 B = I m ; i j with d G (u i,u j ) = 2 Therefore (ij) th block of [A 2 (G) I m ]+[I n A 2 (H)] = (ij) th block of A(G 2 H) Thus, [A 2 (G) I m ]+[I n A 2 (H)] = A(G 2 H) 3 Spectrum of G H In this section, first we recall the spectrum of the graph G We discuss the spectrum of product graphs in terms of spectrum of factor graphs Definition 31 [5] For a matrix A M n n (R), a number λ is an eigenvalue if for some vector X 0, AX = λx The vector X is called an eigenvector corresponding to λ The set of all eigenvalues is the Spectrum of A, and it is denoted by Spec(A), ie, Spec(A) = λ C : λi A = 0} Note that the eigenvalue of G is the eigenvalue of its adjacency matrix A(G) and the spectrum of G is denoted by Spec(A(G)) The following result is known in the usual tensor product A B of matrices A and B as well as in usual tensor product G H of graphs G and H Proposition 32 [9] (i) Let A and B be two square matrices Spec(A B) = λ µ : λ Spec(A), µ Spec(B)} (ii) If G and H are two graphs with n and m vertices respectively, then, Spec(A(G H)) = λµ : λ Spec(A(G)),µ Spec(A(H))} Using Proposition 32(i), we get result similar to (ii) for G 2 H Proposition 33 Let G and H be two graphs with n and m vertices respectively Then, Spec(A(G 2 H)) = λ µ : λ Spec(A 2 (G)), µ Spec(A 2 (H))} 162

6 Proof: Let λ Spec(A(G)) and µ Spec(A(H)) Let X = [x 1,x 2,,x n ] T and Y = [y 1,y 2,,y m ] T be eigenvectorscorrespondingto eigenvalues λ and µ of A 2 (G) and A 2 (H) respectively So, A 2 (G)X = λx and A 2 (H)Y = µy Using kronecker product, we have (A 2 (G)X) (A 2 (H)Y) = (λx) (µy) Therefore (A 2 (G) (A 2 (H))(X Y) = λ µ(x Y) By Proposition 22, we get A(G 2 H)(X Y) = λ µ(x Y) Thus λ µ is an eigenvalue with X Y as an eigenvector of A(G 2 H) The following result is known in the usual cartesian product G H of graphs Proposition 34 [9] (i) Let A and B be two matrices Spec(A B) = λ +µ : λ Spec(A), µ Spec(B)} (ii) If G and H are two graphs with n and m vertices respectively, then, Spec(A(G H)) = λ+µ : λ Spec(A(G)), µ Spec(A(H))} Using Proposition 34 (i), we get result similar to (ii) for G 2 H Proposition 35 Let G and G be two graphs with n and m vertices respectively Then, Spec(A(G 2 H)) = Spec(A 2 (G)) + Spec(A 2 (H)) Proof: Let λ Spec(A 2 (G)) and µ Spec(A 2 (H)) Let X = [x 1,x 2,,x n ] T and Y = [y 1,y 2,,y m ] T be eigenvectorscorrespondingto eigenvalues λ and µ of A 2 (G) and A 2 (H) respectively So, A 2 (G)X = λx and A 2 (H)Y = µy Using kronecker sum, by Proposition 23, we have A 2 (G) A 2 (H) = [A 2 (G) I m ]+[I n A 2 (H)] = A(G 2 H) Therefore A(G 2 H) (X Y) = [A 2 (G) I m ]+[I n A 2 (H)]} (X Y) = [A 2 (G) X I m Y]+[I n X A 2 (H) Y] = [λ X I m Y]+[I n X µ Y] = (λ+µ)(x Y) Thus λ+µ is an eigenvalue with X Y as an eigenvector of A(G 2 H) 4 spectrum of G H in terms of A(G) and A(H) In this section, we find the relation between A 2 (G) and A 2 (G), where A 2 (G) is A(G)A(G), usual matrix multiplication of A(G) with A(G) The degree diagonal matrix D(G) is defined as follows: Definition 412 [9] Let G = (V,E) be a graph with vertex set u 1,u 2,,u n } Then the degree diagonal matrix D(G) = [d ij ] n n is defined as d ij = deg(u i )δ ij = deg(ui ); if i = j Proposition 42 Let G be a simple connected, triangle free and square free graph Then A 2 (G) = A 2 (G) D(G) Proof: Let G be a graph with vertex set u 1,u 2,,u n } Now, A 2 (G) = [c ij ], where c ij = n k=1 a ik a kj ;1 i,j n 163

7 Suppose i j If c ij 0, then for some values of k, a ik and a kj both are non zero, ie, u i is adjacent to u k and u k is adjacent to u j in G Next, if u i u k and u k u j, then u i can not be adjacent with u j in G, as G is a triangle free graph So, u i u k u j gives d G (u i,u j ) = 2 Since G is a square free graph, there is a unique path between u i and u j of length 2 So, for at most one k, a ik and a kj both are non zero Thus c ij = 1, if d G (u i,u j ) = 2 and c ij = 0, otherwise Also, it is known that c ii = deg(u i ) = d ii Thus, A 2 (G) = A 2 (G) D(G) Note that A 2 (G) is a real symmetric binary matrix Remarks 43 (i) Let G be connected but not a triangular free graph Suppose u i u k u j u i is a triangle in G, Then d G (u i,u k ) = 1 = d G (u k,u j ) So, (ij) th entry of A 2 (G) is 1 but (ij) th entry of A 2 (G) is 0, as d G (u i,u j ) 2 Therefore A 2 (G) A 2 (G) D(G) (ii) If G is connected but not a square free graph, then there may be more than one path between two vertices, say u i and u j of length two Then (ij) th entry of A 2 (G) is 2 but (ij) th entry of A 2 (G) is 1 So, A 2 (G) A 2 (G) D(G) Corollary 44 If G ia a k regular graph, triangularfree and square free graph with n vertices, then A 2 (G) = A 2 (G) ki n Consequently, Spec(A 2 (G)) = λ 2 k : λ Spec(A(G))} For example, if G = C n, n 5, then A 2 (C n ) = A 2 (C n ) 2I n In next discussion we fix both the graphs G and H connected, triangular free and square free graphs with n and m vertices respectively Proposition 45 Let G and H be connected graphs (i) A(G 2 H) = [ A 2 (G) A 2 (H) ] [ A 2 (G) D(H) ] [ D(G) A 2 (H) ] +[D(G) D(H)] (ii) A(G 2 H) = [ A 2 (G) I m ] [D(G) Im ]+ [ I n A 2 (H) ] [I n D(H)] Proof: By Proposition 42, A 2 (G) = A 2 (G) D(G) and A 2 (H) = A 2 (H) D(H) (i) By Proposition 22, we have A(G 2 H) = A 2 (G) A 2 (H) So, A(G 2 H) = A 2 (G) A 2 (H) = [A 2 (G) D(G)] [A 2 (H) D(H)] = [ A 2 (G) A 2 (H) ] [ A 2 (G) D(H) ] [ D(G) A 2 (H) ] +[D(G) D(H)] (ii) ByProposition23,A(G 2 H) = [A 2 (G) I m ]+[I n A 2 (H)] Thenbysimilararguments as given in case (i), we get the result Finally, we obtained the spectrum of A(G H) in terms of SpecA(G) and SpecA(H) The following result is known in matrix theory Theorem 46 Let G be k regularand H be s regularconnected graphswith n and m vertices respectively Then Spec(A(G 2 H)) = (λ 2 k) (µ 2 s) : λ Spec(A(G))and µ Spec(A(H))} Proof: Let G and H be regular graphs of n, m vertices with regularity k and s respectively In addition let X be an eigenvector corresponding to eigenvalue λ of G and Y be an eigenvector corresponding to eigenvalue µ of H So, A(G)X = λx and A(H) = µy Also D(G) = ki n and D(H) = si m 164

8 From Proposition 45, we have A(G 2 H) (X Y) = [ A 2 (G) A 2 (H) ] (X Y) [ A 2 (G) D(H) ] (X Y) [ D(G) A 2 (H) ] (X Y) +[D(G) D(H)](X Y) = [ A 2 (G)X A 2 (H)Y ] [ A 2 (G)X D(H)Y ] [ D(G)X A 2 (H)Y ] +[D(G)X D(H)Y] = [ λ 2 X µ 2 Y ] [ λ 2 X si m Y ] [ ki n X µ 2 Y ] +[ki n X si m Y] = [ λ 2 X µ 2 Y ] [ λ 2 X sy ] [ kx µ 2 Y ] +[kx s Y] = [ λ 2 µ 2] (X Y) [ λ 2 s ] (X Y) [ kµ 2] (X Y)+[ks] (X Y) = [ λ 2 µ 2 λ 2 s kµ 2 +k s ] (X Y) = [ (λ 2 k) (µ 2 s) ] (X Y) Thus, Spec(A(G 2 H)) = (λ 2 k)(µ 2 s) : λ Spec(A(G)) and µ Spec(A(H))} Proposition 47 Let G and H be k regular and H be s regular connected graphs of n, m vertices respectively Then Spec(A(G 2 H)) = (λ 2 k) + (µ 2 s) : λ Spec(A(G)),µ Spec(A(H))} Proof: We continue the notations of Theorem 48 Also using Proposition 46, we have A(G 2 H) (X Y) = [A 2 (G) I m ]+[I n A 2 (H)] [D(G) I m ] [I n D(H)]} (X Y) = [A 2 (G) X I m Y]+[I n X A 2 (H) Y] [D(G)X I m Y] [I n X D(H)Y] = [λ 2 X I m Y]+[I n X µ 2 Y] [kx Y] [X sy] = ((λ 2 k)+(µ 2 s))(x Y) Thus, Spec(A(G 2 H)) = (λ 2 k)+(µ 2 s) : λ Spec(A(G)),µ Spec(A(H))} Acknowledgment The research is supported by the SAP programmed to Department of Mathematics, S P University, Vallabh Vidyanagr, by UGC References [1] U P Acharya and H S Mehta, 2- Cartesian Product of Special Graphs, Int J of Math and Soft Comp, Vol4, No1, (2014), [2] U P Acharya and H S Mehta, 2 - tensor Product of special Graphs, Int J of Math and Scietific Comp, Vol4(1), (2014), [3] U P Acharya and H S Mehta, Generalized Cartesian product of Graphs, Int J of Math and Scietific Comp, Vol5(1), (2015), 4-7 [4] S K Ayyaswamy, S Balachandran and I Gutman, On Second Stage Spectrum and Energy of a Graph, Kragujevac J of Math 34, (2010), [5] R Balakrishnan and K Ranganathan, A Text book of Graph Theory, Universitext, Second Edition, Springer, New York, 2012 [6] R B Bapat: Graphs and Matrices, Second Edition, Springer-Verlag, London and Hindistan Book Agency, India, 2011 [7] C Godsil and G F Royle: Algebraic Graph Theory, Springer-Verlag, New York, 2001 [8] Richard Hammack, Wilfried Imrich and Sandi Klavzar, Hand book of product Graphs, CRC Press, Taylor & Francis Group, 2 nd edition, New York, 2011 [9] D M Cvetkovic, M Doob and H Sachs, Spectra of Graphs, Academic Press, New York,