Generation of Strongly Regular Graphs From Normalized Hadamard Matrices

7/29/2019 Generation of Strongly Regular Graphs From Normalized Hadamard Matrices

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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 2, ISSUE 2, FEBRUARY 2013 ISSN 2277-8616

212IJSTR2013www.ijstr.org

Figure 11- Latin square of order 4

Figure 12- Latin square of order 8

In this paper we have proposed an algorithm which can beused to generate strongly regular graphs from normalizedHadamard matrices of order for .

2METHODOLOGYThe proposed method constructs a graphfrom a normalized Hadamard matrix. Since, the finiteprojective plane is known to be the smallest design ,thismethod is developed to begin with the normalizedHadamard matrix of order 4.

2.1Mathematical Background Of The ProposedMethod

For any integer the normalized Hadamard matrix oforder can be constructed using the Sylvestersconstruction. Hence, the Sylvester construction cited infigure 3 can be applied to the smallest order Hadamardmatrices defined in figure 1 to obtain the Hadamard matrix

of order .For an example, The normalized Hadamardmatrix of order 4 can be constructed using asfollows:

Figure 13The normalized Hadamard matrix of order 4constructed by Sylvester construction

In the next step label each column vector as for.For the above example, there are four column

vectors .

Figure 14- The column vectors of the matrix

Then, multiply each column vector by its transpose andlabel the resulting matrices as , for .For theabove example, each s can be constructed as follows.

Using the first column vector , the construction of the firsblock matrix is as follows:

Figure 15-The construction of the first block matrix

Then, applying the above matrix multiplication method tothe column vector gives the next block matrix .

Figure 16-The second block matrix

Similarly, the matrix operation on the third column vectoryields the next block

Figure 17-The third block matrix

Since the illustration uses the matrix , there are fouscolumn vectors and the last one being . Hence matrixproduct on gives the required last block .

Figure 18-The last block matrix

Then consider the symmetric Latin square with entrieswith constant diagonal 1 and replace each by. For the above example, the symmetric Latin square o

four elements is required. The resulting Latin square will beas follows,

Figure 19-The Latin square of order 4


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Hence, by replacing each element with the correspondingmatrix, following large matrix of order 16 can be

obtained.

Figure 20-The constructed matrix which is replaced by theblock matrices

Figure 21-The matrix obtained by substituting the blockmatrices

Then, replacing 1 to 0 and -1 to 1 yields to the adjacencymatrix of the strongly regular graph obtained from thenormalized Hadamard matrix of order .For the Hadamardmatrix of order 4 we have the following adjacency matrix.

Figure 22-The adjacency matrix of the strongly regulargraph obtained from

The same method is applicable for the other normalizedHadamard matrices of order for .

2.2 Implementation Of The Proposed MethodThis algorithm was implemented using C++ programminglanguage. The first task of this computer program is togenerate a normalized Hadamard matrix with a specifiedorder. This task is achieved by implementing the kronecker

product of matrices starting from the normalized Hadamardmatrix of order two mentioned above. User can input thevalue of the required Hadamard matrix of order for.Depending on the user input, the program generates the

relevant normalized Hadamard matrix and stores it in anexternal file (File 1) to reduce the burden of holding largematrices in the computer RAM (Random Access Memory)In the same way, the program creates the relevant

symmetric Latin Square and stores it in another external file(File 2).

Next the program reads column vectors (simple c'sex: ,....) from the normalized Hadamard matrix storedin that external file (File 1) and execute the product of itstranspose to get the resulting matrices (capital C's ex ..). In the latter part of the computer program the generatedmatrices ..etc will be substituted to the relevanlocations of the above generated Latin square. Theresulting matrix will then be saved in another external file(File 3) Finally, it will automatically replace with and

with and results the required adjacency matrix of thestrongly regular graph will be generated and this resulting

adjacency matrix will be saved as another external file (File4). This File 4 will finally be used to draw the stronglyregular graph on the computer display by drawing linesbetween all the adjacent vertices.

3 RESULTS AND DISCUSSIONIt is easy to observe that, each block that are constructedby the operation for are symmetric. That is

for .Since the normalized Hadamardmatrices have been used, in every construction the firstcolumn vector will be a column matrix with all entries beingone. Hence the first block in each construction is a matrixof order whose entries are equal to one. That is,

,where is a matrix of order with all entries equal toone. Further, it can be seen that fo

. The implementation of the above methodwas tested for several normalized Hadamard matrices oorder for . For the program, the following Hardwareand software were used.

Hardware:System Model: Inspiron N5110Computer: Intel(R) Core(TM) i7-2670QM CPU @ 2.20GHz(8 CPUs), ~2.2 GHzMemory: 8192 MB RAM

Software:Operating System: Windows 7 Home Basic 64-bit (6.1Build 7601)

IDE: Turbo C++; Version 3.0; Copyright(c) 1990, 1992 byBorland International, Inc;DosBox 0.74

Here, in the program, the user is free to enter the order othe Hadamard matrix, coordinates of the graph radiusof the circle which is used to draw the graph and the colouof the graph.


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Figure 23-The user input interface

Using the Hadamard matrix of order 4,we can obtain SRG(16,6,2,2), that is the graph consists of 16 points for whicheach vertex has the degree 6 and each pair of adjacencyvertices appear exactly twice and each non-adjacency pairof vertices have 2 common neighbours.Hence the requiredRegular graph is drawn as follows.

Figure 24-The strongly regular graph obtained fromThe corresponding adjacency matrix of the above graph isshown below.

Figure 25-The adjacency matrix of the graph obtained from

Moreover, the regular graph (64,28,12,12) which can beobtained from the Hadamard matrix of order 8 is shownbelow.

Figure 26- The regular graph obtained from

Thus, from this method, one can obtain the strongly regulagraph of order from the normalized Hadamard matrixof order .At the same time, one can get the adjacencymatrix of the relavant graph.

4 CONCLUSIONThe adjacency matrix of each graph is symmetric and thediagonal elements are all zeros and satisfies the properties

and ,where denotesthe identity matrix of order and denotes the matrix withall ones. Hence, all the graphs generated from this methodare strongly regular graphs with , ,

.

5 REFERENCES[1] Wallis, J. S. (1975). On Hadamard matrices. Journal o

Combinatorial Theory , 149-164.

[2]Craigen, R. (1996). Hadamard matrices and designs. In J. HC. J. Colbourn, CRC Handbook of Combinatorial Designs (pp370-377). CRC Press.

[3]Graybill, F. A. (1983). Matrices, with Applications in StatisticsBelmont: Wadsworth International Group.

[4]J.J.Sylvester. (1867). Thoughts on orthogonamatrices,simultaneous sign successions and tessellatedpavements in two or more colours with applications to Newtonsrule,ornamental tile work and the theory of numbers. 461-475Phil.Mag .

[5]K.J.Horadam. (2007). Hadamard matrices and generalizations. In Hadamard matrices and their applications (pp. 10-12)New Jersy: Princeton University Press.

[6]Jennifer Seberry, M. Y. (1992). Hadamard matricessequences, and block designs. In D. J.H.Dinitz, Contemporarydesign theory-A collection of surveys (pp. 431-560). JohnWiley and sons.

[7]Harju, T. (2011). Lecture notes on graph theory.

[8]David Joyner, M. V. Algorithmic graph theory.

[9]Diestel, R. (1991). Graph Theory, Graduate Texts inMathematics. Berlin: Springer Verlag.

[10]Prajapati, M. C. (n.d.). Distance In Graph Theory And Its

Application. International Journal of Advanced EngineeringTechnology .

[11]West, D. (2000). Introduction to Graph Theory. Prentice Hall.

[12]Cameron, P. .. (2001). Strongly Regular Graphs. LondonUniversity of London.

[13]Brouwer, A. E. (1984). Strongly Regular Graphs and PartiaGeometries. Enumeration and design:Conference onCombinatorics (pp. 85-122). Waterloo: Academic press.


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215IJSTR2013www ijstr org

[14]Cormen, T. H. (2001). Representations of graphs. InIntroduction to Algorithms (pp. 527-531). MIT Press andMcGraw-Hill.

[15]R.C.Bose. (1963). Strongly Regular graphs,partial geometriesand partially balanced designs. Pacific Journal ofMathematics.

[16]G.H. J. van Rees. (2009). More greedy defining sets in Latinsquares. Australasian Journal Of Combinatorics , 183-198.

[17]Charles .J. Colbourn, (2007). The Latin square . In Handookof Combinatorial designs (pp. 135-137). CRC Press.

A.A.C.A.Jayathilake is anundergraduate at the University ofPeradeniya, Sri Lanka. She isfollowing a BSc (Special) degree inMathematics, University ofPeradeniya, Sri Lanka (2013). Herresearch interests areCombinatorics, Graph theory,

Differential equations and design theory.

A.A.I.Perera is a Professor inMathematics at the Department ofMathematics , University ofPeradeniya, Sri Lanka. His researchinterests are Combinatorics,Graphtheory,Design Theory and GroupTheory.

M. A. Pathum Chamikara isworking as a research assistant atthe Post Graduate Institute ofScience (PGIS), University ofPeradeniya, Sri Lanka. He receivedhis BSc (Special) degree inComputer Science, University ofPeradeniya, Sri Lanka (2010). His

research interests include Crime analysis, GIS (GeographicInformation Systems), image processing, computationalmathematics, computer vision and artificial intelligence.

Generation of Strongly Regular Graphs From Normalized Hadamard Matrices

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