Malaya Journal of Matematik, Vol. 8, No. 3, 1321-1327, 2020 https://doi.org/10.26637/MJM0803/0103 A note on Frobenius inner product and the m-distance matrices of a tree P.A. Asharaf1* and Bindhu K. Thomas2 Abstract The m-distance matrix Dm of a simple connected undirected graph has an important role in computing the distance matrix D of the graph from the powers of the adjacency matrix using Hadamard product. This paper shows that for an undirected tree T with diameter d, fD0:D1;:::;Ddg is an orthogonal basis for the space spanned by the binary equivalent matrices of the first d + 1 powers of the adjacency matrix of T and it gives an invertible conversion matrix for finding the m-distance matrix of T using Frobenius inner product on matrices. Keywords Adjacency matrix, Distance matrix, Binary matrix , Diameter, Hadamard product, m-distance matrix , Frobenius inner product, Frobenius norm. AMS Subject Classification 05C12, 05C40, 05C50, 05C62, 05B20, 46C05, 15A60. 1Mathematics Research Centre, Mary Matha Arts and Science College, Mananthavady-670645, Kerala, India. 2Department of Mathematics, Mary Matha Arts and Science College, Mananthavady-670645, Kerala, India. *Corresponding author: 1 [email protected]; [email protected] Article History: Received 12 April 2020; Accepted 19 August 2020 ©2020 MJM. Contents Definition 1.2. ([3]) The diameter of a graph G is the maxi- mum distance between any two vertices of G and it is denoted 1 Introduction......................................1321 by Diam(G). 2 Preliminaries.....................................1322 Definition 1.3. (Hadamard product [4]) Consider the vector m× n 3 Main Resuls......................................1322 space R of all m × n real matrices over the real field m× n 4 Illustration.......................................1325 R. For P; F 2 R , the Hadamard product ◦ is a binary operation on m× n defined by, 5 Conclusion.......................................1327 R References.......................................1327 (P ◦ F)i j := (P)i j(F)i j; 8 i; j: Let B = f0;1g and let Bm×n denote the set of all binary matri- 1. Introduction m×n ces in R . Consider a simple, connected, undirected graph G = (V;E) Then for P; F 2 Bm× n, of order n with vertex set V and edge set E throughout this 1; if pi j = 1 and fi j = 1 paper unless otherwise specified. Let AG be the n × n adja- (P ◦ F)i j = th m th 0; otherwise cency matrix [1] of G. Then i j entry of AG (m power of AG), represent the number of walks of length m between the Definition 1.4. (Frobenius inner product [5]) Consider the vertices v and v of G. n× n i j vector space R over the field R. Then the Frobenius inner n× n n× n product h;iF : × ! is defined by, Definition 1.1. ([2]) The distance matrix, D = (di j) of G is R R R defined as, n n n n hP;Qi = p q = (P ◦ Q) = Tr(PQT ); F ∑ ∑ i j i j ∑ ∑ i j d(vi;v j) , if i 6= j i=1 j=1 i=1 j=1 di j = 0 , if i = j n×n where P;Q 2 R . The Frobenius norm k · k induced by this n×n 1 n×n Where, d(vi;v j) is the distance between the vertices vi and v j. inner product on R is, kPkF = (hP;PiF ) 2 , P 2 R . Then A note on Frobenius inner product and the m-distance matrices of a tree — 1322/1327 n×n the metric dF induced by this norm on R is dF (P;Q) = Remark 2.3. (i)D m 2 Bn×n, 8 m = 0;1;2;::: n×n kP − QkF ,P;Q 2 R : (ii)D 0 = In 1 1 T 2 Remark 1.5. (i) kPkF = (hP;PiF ) 2 = Tr(PP ) , P 2 (iii)D + D + ::: + D = n×n 0 1 d R 21 1 ::: 13 (ii) Let X;Y 2 1×n. Then hX;Yi = n x y and R F ∑i=1 i i 61 1 ::: 17 6 7 1 6::::::7 n ! 2 Jn = 6 7 = The matrix of ones 2 6::::::7 d f (X;Y) = kX −YkF = ∑(xi − yi) 6::::::7 i=1 4 5 1 1 ::: 1 which is the Euclidean distance between the vectors X and Y of n. Diam(G) = Maxfm : D 6= 0g R (iv) m m (iii) For P 2 Bn×n, 0; if i 6= j 1 ! (v)D i ◦ D j = ; for 0 ≤ i; j ≤ d: n n 2 Di; if i = j d (P;0) = kP − 0k = kPk = p2 n×n F F F ∑ ∑ i j i.e., fD0;D1;:::;Ddg is an orthogonal subset R i=1 j=1 with respect to the binary operation ◦: 2 Then (dF (P;0)) represents the number of 1’s in P. Theorem 2.4. ([6]) Let AG be the adjacency matrix of G = For P;Q 2 Bn×n, (V;E) with diameter d. Then b2 = fD0;D1;:::;Ddg is a Lin- n×n 1 early independent subset of the vector space R . n n ! 2 2 dF (P;Q) = kP − QkF = ∑ ∑(pi j − qi j) Theorem 2.5. ([6]) Let AG be the adjacency matrix of G = i=1 j=1 (m) (V;E) with diameter d. Then for 1 ≤ m ≤ d; Dm = AG − m−1 (m) (s) 2 d ∑ A ◦ A , where Dm is the m-distance matrix of Then (dF (P;Q)) represent the number of non identical entries s=0 G G in P and Q and the number of identical entries in P and Q is G. 2 2 n − (dF (P;Q)) 3. Main Resuls 2. Preliminaries Theorem 3.1. Let AG be the adjacency matrix of G = (V;E). Definition 2.1. ([6]) The function d : R ! B = f0;1g is de- Let d = diam(G): Then for 0 ≤ K; m ≤ d; fined by (m) Dm; if K = m Dk ◦ A = 0; a ≤ 0 G 0; if m < K: d(a) := ;a 2 1; otherwise R Proof. Case(i): K = m m×n Also, d : R ! Bm×n is defined by d(D) = (d(D))i j := (m) (m) m×n m Dm ◦ AG = (Dm)i j(AG )i j (d(di j)); 8 i; j and D 2 R : Then d(AG) 2 Bn×n, 8 m 2 i j f0;1;2;:::g and it is the equivalent binary matrix representa- (m) 1; if (D ) = 1 and (A ) = 1 Am A(m) = (Am) = m i j G i j tion of G. Let G d G . Then 0; otherwise 8 (m) 1; if there exist a walk of length m 1; if d(vi;v j) = m and (A )i j = 1 (m) < = G (AG )i j = from vi to v j in G 0; otherwise. : 0; otherwise (m) But d(vi;v j) = m ) (AG )i j = 1 Also, for C;F 2 Bm×n, d(C ◦ F) = d(C) ◦ d(F): (m) (m) Dm ◦ AG = (Dm)i j(AG )i j Definition 2.2. (m-distance matrix [6]) Consider the graph ) i j G = (V;E) with n vertices. Then the m- distance matrix D m 1; if d(v ;v ) = m of G is an n × n symmetric binary matrix defined by = i j = (D ) 0; otherwise m i j 1; if d(vi;v j) = m (m) (Dm)i j := 0; otherwise That is, (Dm ◦ AG )i j = Dm: Case(ii): K > m (m) (m) where, d(vi;v j) is the distance between the vertices vi and v j. (Dk ◦ AG )i j = (Dk)i j(AG )i j 1322 A note on Frobenius inner product and the m-distance matrices of a tree — 1323/1327 If (Dk)i j = 1; then d(vi;v j) = K: Lemma 3.4. Let G = (V;E) be a connected undirected graph That is, if there exist a shortest path of length K between of order n(n > 1). Let di j denote the distance between vi and vi and v j: But then this path cannot be traversed by any of v j. Then there always exist at least one walk between vi and (m) v j of length di j + 2r; 8 r = 0;1;2;:::. That is, the walk between vi and v j of length< K: Thus (AG )i j = 0: (* K > m) (di j+2r) (m) AG = 1;8r = 0;1;2;:::: ) (DK)i j(AG )i j = 0 i j (m) If (DK)i j = 0; then also (DK)i j(AG )i j = 0. Proof. Since G is a connected undirected graph, there exist (m) Therefore, (DK ◦ AG )i j = 0; 8 i; j; 8 K > m: at least one path in between vi and v j. Let Pi j : v0(= vi) − (m) v1 − v2 − ::: − vd −1 − vd (= v j) denote the shortest path ie., DK ◦ AG = 0 if K > m: Hence i j i j between vi and v j of length di j. If we traverse back and forth once along the last edge in P , then we get a walk v (= (m) Dm; if K = m i j 0 DK ◦ A = ; for K ≤ d: G 0; if m < K vi) − v1 − v2 − ::: − vdi j−1 − vdi j (= v j) − vdi j−1 − vdi j (= v j) of length di j + 2 between vi and v j. Also If we traverse back and forth twice along the last edge in Pi j, then we get a walk v0(= vi)−v1 −v2 −:::−vdi j−1 −vdi j −vdi j−1 −vdi j −vdi j−1 − Theorem 3.2.