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  1. This is a talk on The Magnificent Matrix and its next Generation structures

  2. Delivered in the Spring Workshop on Combinatorics and Graph Theory, 2006Held atCenter for CombinatoricsNankai UniversityTianjinPeoples’ Republic of Chinaon April 21, 2006

  3. By Prof. R.N.Mohan Sir CRR Institute of Mathematics Eluru-534007, AP, India Andhra University ---------- Visiting Professor TWAS-UNESCO Associate member Center for Combinatorics Nankai University, Tianjin, PR China

  4. Magnificent Matrixotherwise called as • Mn-Matrix is a square matrix obtained by: (aij) = (di x dh dj) mod n, by suitably defining di , dh , dj , x in different ways. For example: • 1.1+(i-1)(j-1) mod n (for n is a prime) • 2. (i.j) mod n (for n or n+1,is a prime) • 3 (i+j) mod n (for n is a positive integer) • Still there are so many ways to explore

  5. The three types mentioned here are combinatorially equivalent And each is useful in its own way for the construction of many : Combinatorial Configurations

  6. The combinatorial configurations mainly are • Balanced Incomplete Block (BIB) Designs • Partially Balanced Incomplete Block (PBIB) Designs • Symmetric BIB and PBIB designs • Graphs • Latin squares, orthogonal arrays, sub arrangements, Youden squares etc.

  7. The Mn-Matrices • Gives rise to Mn-Graphs, defined as • If given an Mn-matrix: • Ck’s be its columns • aij’s be its elements • let V1 = {Ck}, V2 ={aij} be the vertex-sets • An edge is αijk iff aij is in Ck. • This gives the Mn-graph (V1, V2, αijk)

  8. LDPC code • By using the pattern of Mn-matrix aij = 1+(i-1)(j-1) mod n • Bane Vasic and Ivan of Arizona, USA Constructed Low-density Parity Check (LDPC) Codes

  9. These Mn-matrices • Have been used in the construction of these BIB and PBIB designs • A BIB designs, is an arrangement in which • v elements are arranged in b blocks, • each element is coming in r blocks • and each block is having k elements • and each pair of elements is coming in λ blocks.

  10. If λ is not constant • Then they are called as: Partially balanced incomplete block designs • If v = b and r = k then the design is called • Symmetric design

  11. These designs are used In Communication & Networking systems by Charles Colbourn, Dinitz and Stinson. Jointly and independently, and by many others also

  12. specifically Mn-matrices • Gave the method of construction of • μ-resolvable and • Affine μ-resolvable BIB and PBIB designs

  13. Affine Resolvability, Resolvability • If the b blocks are grouped in to t sets of m blocks each then the design is said to beResolvable • If the blocks of the same set have treatments in common • If the blocks of different sets have treatments in common then they are called as affine resolvable designs

  14. Application • Thus when blocks are grouped into parallel classes then the resolvability exist in a design, limited block intersection leads to affine nature. • These classes are called resolution classes • If the set of m messages assigned to a particular user forms a parallel class or resolution class

  15. Then comes the next generation • These Mn-matrices lead to the construction of Three types of M-matrices(The next Generation) • namely: • Type I is with1+(i-1)(j-1) mod n,Prime • Type II is with (i.j) mod n (n+1 prime) • Type III is with (i+j) mod n (n is an integer) • And their corresponding M-graphs

  16. Those are defined as • M-matrix of Type I • Definition. When n is a prime, • consider the matrix of order n obtained by the equation • Mn = (aij), where • aij = 1 + (i-1)(j-1) mod n, when n is prime where i, j = 1, 2,.., n

  17. M-matrix of Type I • In the resulting matrix • retain 1 as it is • substitute -1’s for odd numbers • substitute +1’s for even numbers. • This gives M-matrix of Type I. • This is a symmetric n x n matrix. • Roles of +1 and -1 can be inter-changed

  18. Hadamard matrix • A matrix H having • All +1’s in the first row and first column • HH′= nIn • It is an orthogonal matrix • This is an important matrix having many applications

  19. Resemblances and Differencesbetween M-Matrix & Hadamard Matrix. • Both have +1’s in the first row and first column • Both consist of +1 or -1 only • Row sum in (M) is 1 and in (H) is zero • (M) Exists for all primes, (H) exists for n =2 or 0 mod 4 • Both useful for the constructions of codes, graphs, and designs, and Sequences and array sequences • (M) is Non-orthogonal, (H) is orthogonal,

  20. Properties of M-matrix of Type I • in each row and column, the number of +1’s is (n+1)/2 • and the number of -1’s is (n-1)/2.

  21. The orthogonal numbers are • the orthogonal number between any two rows • is given by 4k+2-n, • where k is the number of +1’s in the selected set

  22. The orthogonal numbers are defined by • The formula

  23. A property

  24. Sum • The sum of the orthogonal numbers is given by (n+1)/2

  25. Because it is given by • By the formula

  26. Here is an open problem • Do all orthogonal numbers as per the formula exist in a matrix concerned now? • For example when n = 11, the orthogonal numbers are -9, -5, -1, 3, 7, 11. • But -5 and 7 do not exist. They are called missing orthogonal numbers.

  27. Why they miss???? We consider the sum of all orthogonal numbers including these missing numbers

  28. Determinant • Given an M-matrix of Type I |M| = - 4 if n = 3 = 0 if n ≥ 5, In an (1,-1)-matrix, when the determinant is maximum then it is called as Hadamard Matrix

  29. SPBIB design • The existence of an M-matrix of order n, where n is a prime, implies the • existence of an SPBIB design with parameters • v = b = n-1, r = k = (n-1)/2, • λi vary from 0 to (n-3)/2.

  30. Graph • The existence of an M-matrix of Type I • implies the existence of • A regular bipartite graph V = 2n (V1= n, V2=n), E = 2n valence is (n-1)/2

  31. Example • From Mn = [aij], • where aij = 1 + (i-1)(j-1) mod n, • n is a prime • i, j = 1, 2, 3,4,5.

  32. Mn-matrix • Is given by

  33. M-matrixof Type I • Is given by

  34. SPBIB design • Is given by • v = 4 = b, r = k = 2, λ1 = 1, λ2 = 0. • The solution is 1 3 1 2 3 4 2 4

  35. M-Graph (next generation) • And the graph is

  36. Usable • These types of graphs form a new family of graphs, which are highly usable in • routing problems of • salesmen, transportation or • Communication and network systems

  37. For n = 11 • We get an M-graph as

  38. M-matrix Type II • When n + 1 is a prime, • Take aij = (i j) mod (n+1), i, j = 1,2,...,n

  39. Orthogonal numbers • orthogonal number between any two rows Is given by • g = 4k-n where k is the number of 1’s in the selected set.

  40. The sum of orthogonal numbers • Is given by

  41. SPBIB design • The existence of an M-matrix of type II, implies the existence of an SPBIB design with parameters • v = n= b, r =k = n/2, • λi values vary from 0 to (n-3)/2.

  42. Graph • The existence of an M-matrix of type II, • implies the existence of • a Regular Bipartite Graph.

  43. For n+1 = 7 • The SPBIB design, is given by • 1 4 1 2 1 2 • 3 5 4 3 2 4 • 5 6 5 6 3 6 • where as v = b = 6, r = k = 3, λ1= 2, λ2 = 1, λ3 = 0, n1 = 2, n2 = 2, n3 = 1

  44. M-Graph • Its regular bipartite graph is as follows:

  45. These M-graphs give A new family of fault-tolerant M-networks • We will show some of its features

  46. The main features of M-networks • The maximum diameter of the M-network is found to be 4 independent of the network size. • M-networks out-perform other known regular networks in terms of throughput and delay. • exhibit higher degree of fault-tolerance • as these graphs have good connectivity

  47. Reliability • they provide a reliable communication system • These networks are found to be denser than many known multiprocessor architectures • such as mesh, star, ring, the hypercube

  48. Lastly another application There are n nodes in the network, and they are to be inter-connected by using Buses. A Bus is a communication device, which connects two or more nodes and provides a direct connection between any pair of nodes on the bus.

  49. M-matrix of Type III • This matrix is obtained by (i+j) mod n • When n is an integer odd or even • Not necessarily prime

  50. In similar way In the resulting matrix substitute 1 for even numbers and -1 for odd numbers and also for 1, ( or 1 for odd numbers keeping the 1 in the matrix as 1 itself and -1 for even numbers).