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**This is a talk on**The Magnificent Matrix and its next Generation structures**Delivered in the Spring Workshop on Combinatorics and Graph**Theory, 2006Held atCenter for CombinatoricsNankai UniversityTianjinPeoples’ Republic of Chinaon April 21, 2006**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**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**The three types mentioned here are combinatorially**equivalent And each is useful in its own way for the construction of many : Combinatorial Configurations**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.**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)**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**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.**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**These designs are used**In Communication & Networking systems by Charles Colbourn, Dinitz and Stinson. Jointly and independently, and by many others also**specifically Mn-matrices**• Gave the method of construction of • μ-resolvable and • Affine μ-resolvable BIB and PBIB designs**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**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**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**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**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**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**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,**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.**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**The orthogonal numbers are defined by**• The formula**Sum**• The sum of the orthogonal numbers is given by (n+1)/2**Because it is given by**• By the formula**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.**Why they miss????**We consider the sum of all orthogonal numbers including these missing numbers**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**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.**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**Example**• From Mn = [aij], • where aij = 1 + (i-1)(j-1) mod n, • n is a prime • i, j = 1, 2, 3,4,5.**Mn-matrix**• Is given by**M-matrixof Type I**• Is given by**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**M-Graph (next generation)**• And the graph is**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**For n = 11**• We get an M-graph as**M-matrix Type II**• When n + 1 is a prime, • Take aij = (i j) mod (n+1), i, j = 1,2,...,n**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.**The sum of orthogonal numbers**• Is given by**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.**Graph**• The existence of an M-matrix of type II, • implies the existence of • a Regular Bipartite Graph.**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**M-Graph**• Its regular bipartite graph is as follows:**These M-graphs give A new family of fault-tolerant**M-networks • We will show some of its features**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**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**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.**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**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).