Application of Tikhonov Regularization in
Generalized Inverse of Adjacency Matrix of
Undirected Graph

Paul Ryan A. Longhas; Alsafat M. Abdul

doi:https://doi.org/10.14445/22315373/IJMTT-V68I2P501

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 68 | Issue 2 | Year 2022 | Article Id. IJMTT-V68I2P501 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I2P501

Application of Tikhonov Regularization in Generalized Inverse of Adjacency Matrix of Undirected Graph

Paul Ryan A. Longhas, Alsafat M. Abdul

Abstract

In this article, we found the Moore-Penrose generalized inverse of adjacency matrix of an undirected graph, explicitly. We proved that the matrix
𝑅𝜆 = [𝑟𝑖𝑗] is nonsingular where for 𝑖 ≠ 𝑗 and, we proved that
The proof of the main result was based on the Tikhonov regularization.

Keywords

Moore-Penrose generalized inverse, Adjacency matrix, Tikhonov regularization, Undirected graph, Nonsingularity.

References

[1] Groetsch C.W, Generalized Inverses of Linear Operators: Representation and Approximation, Marcel Dekker, New York. (1970).
[2] Israel A. and Greville T., Generalized Inverses: Theory and Applications (2nd ed.). New York, NY: Springer. (2003).
[3] Campbell S.L. and Meyer C.D., Generalized Inverses of Linear Transformations. (1991).
[4] Diestel R., Graph Theory, Springer-Verlag Heidelberg, New York. (2005).
[5] Chartrand, G. Introductory Graph Theory. New York: Dover. (1985) 218.
[6] Skiena S, Adjacency Matrices. §3.1.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, (1990) 81-85.
[7] West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall. (2000) 6-9.
[8] Biggs, Norman, Algebraic Graph Theory, Cambridge Mathematical Library (2nd ed.), Cambridge University Press, Definition 2.1. (1993) 7.
[9] Lawson, C. and Hanson, R. Solving Least Squares Problems. Englewood Cliffs, NJ: Prentice-Hall. (1974).
[10] Penrose R, A Generalized Inverse for Matrices. Proc. Cambridge Phil. Soc. 51 (1955) 406-413.
[11] Rao C. R, and Mitra S. K, Generalized Inverse of Matrices and Its Applications. New York: Wiley. (1971).
[12] Muntaha Khudair Abbas, Properties on Some Types of Generalized Inverses of Matrices-A Review, International Journal of Mathematics Trends and Technology. 65(1) (2019) 66-72.
[13] Tikhonov, Andrey Nikolayevich, On the stability of inverse problems, Doklady Akademii Nauk SSSR. 39(5) (1943) 195–198.
[14] Tikhonov, A. N.. On the solution of ill-posed problems and the regularization method. Doklady Akademii Nauk SSSR. 151 501–504. Translated in "Solution of incorrectly formulated problems and the regularization method", Soviet Mathematics. 4 (1963) 1035–1038.
[15] Tikhonov A. N, V. Y. Arsenin, Solution of Ill-posed Problems, Washington: Winston & Sons. (1977).
[16] Tikhonov, Andrey Nikolayevich, Goncharsky A, Stepanov V. V, Yagola Anatolij Grigorevic, Numerical Methods for the Solution of Ill-Posed Problems. Netherlands: Springer Netherlands. (1995).
[17] Tikhonov, Andrey Nikolaevich, Leonov, Aleksandr S, Yagola, Anatolij Grigorevic. Nonlinear ill-posed problems. London: Chapman & Hall. (1998).
[18] Banerjee, Sudipto, Roy, Anindya, Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC. (2014).
[19] Trefethen, Lloyd N, Bau III, David. Numerical linear algebra. Philadelphia: Society for Industrial and Applied Mathematics. (1997).
[20] Faudree Ralph, Flandrin Evelyne, Ryjáček Zdeněk, Claw-free graphs — A survey, Discrete Mathematics. 164 (1–3) (1997) 87–147.
[21] Chudnovsky, Maria; Seymour, Paul, The structure of claw-free graphs, Surveys in combinatorics 2005 (PDF), London Math. Soc. Lecture Note Ser., Cambridge: Cambridge Univ. Press, MR 2187738. 327 (2005) 153–171.
[22] Knuth Donald E, Two thousand years of combinatorics, in Wilson, Robin; Watkins, John J. (eds.), Combinatorics: Ancient and Modern, Oxford University Press, (2013) 7–37.
[23] Gries David, Schneider Fred B, A Logical Approach to Discrete Math, Springer-Verlag, 327 (1993) 436.
[24] Meyer Jr, Carl D, Generalized inverses and ranks of block matrices. SIAM J. Appl. Math. 25(4) (1973) 597–602.
[25] Meyer Jr, Carl D, Generalized inversion of modified matrices. SIAM J. Appl. Math. 24(3) (1973) 315–23.

Citation :

Paul Ryan A. Longhas, Alsafat M. Abdul, "Application of Tikhonov Regularization in Generalized Inverse of Adjacency Matrix of Undirected Graph," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 2, pp. 1-6, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I2P501