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A Time-Based Solution for the Graph Isomorphism Problem

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Abstract

This paper is devoted to the problem of isomorphism in graphs and proposes a method based on time response of a differential equation. First it is shown that the solution of a differential equation obtaining from a Laplacian matrix can be used as an index and the proof is presented. Then a search algorithm is proposed to find out that the two graphs are isomorphic and there is a permutation matrix describing relations between the graphs. The search algorithm depends on eigenvalues of the Laplacian matrix. For a Laplacian matrix with repeated eigenvalues, Greshgorin theorem is used to convert it to a matrix with non-repeated eigenvalues. This is performed by adding loops to vertices, so that they have separated Greshgorin bands. Then the time response of the differential equation is checked. The proposed method is performed on co-spectral graphs and results are described.

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References

  1. R.E.L. Aldred, M.N. Ellingham, R.L. Hemminger, P. Jipsen, P3-isomorphisms for graphs. J. Graph Theory 26(1), 35–51 (1997)

    Article  MathSciNet  Google Scholar 

  2. A. Atserias, E. Maneva, Graph isomorphism, Sherali-Adams relaxations and expressibility in counting logics, in Electronic Colloquium on Computational Complexity, vol. 77 (2011), pp. 1–34

  3. L. Babel, Recognition and isomorphism of tree-like P4-connected graphs. Discrete Appl. Math. 99(1–3), 295–315 (2000)

    Article  MathSciNet  Google Scholar 

  4. A. Dharwadker, J. T. Tevet, The graph isomorphism algorithm, in Proceedings of the structure semiotics research group s.e.r.r. eurouniversity, TALLINN (2009), pp. 1–30

  5. E. Dobson, I. Kovacs, S. Miklavic, The isomorphism problem for window graphs. Discrete Math. 323, 7–13 (2014)

    Article  MathSciNet  Google Scholar 

  6. J.A. Fill, D.E. Fishkind, E.R. Scheinerman, Affine isomorphism for partially ordered sets. Order 15(2), 183–193 (1999)

    Article  MathSciNet  Google Scholar 

  7. K. Fukuda, M. Nakamori, Graph isomorphism algorithm by perfect matching, in IFIP Conference on System Modelling and Optimization, vol. 130 (2003), pp. 229–238

  8. M. Furst, J. Hopcroft, E. Luks, Polynomial-time algorithm for permutation groups, in 21st Annual Symposium on Foundations of Computer Science (1980), pp. 36–41

  9. Z.H. Guan, F.L. Sun, Y.W. Wang, T. Li, Finite time consensus for leader-following second-order multi-agent networks. IEEE Trans. Circuits Syst. I 59(11), 2646–2654 (2012)

    Article  MathSciNet  Google Scholar 

  10. W.H. Haemers, E. Spence, Enumeration of co-spectral graphs. Eur. J. Combin. 25(2), 199–211 (2004)

    Article  Google Scholar 

  11. D.S. Johnson, The NP-completeness column. ACM Trans. Algorithms 1(1), 160–176 (2005)

    Article  MathSciNet  Google Scholar 

  12. S. Kijima, Y. Otachi, T. Saitoh, T. Uno, Subgraph isomorphism in graph classes. Discrete Math. 312(21), 3164–3173 (2012)

    Article  MathSciNet  Google Scholar 

  13. Z. Li, Z. Duan, G. Chen, L. Huang, Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE Trans. Circuits Syst. I 57(1), 213–224 (2010)

    Article  MathSciNet  Google Scholar 

  14. E.M. Luks, Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci. 25(1), 42–65 (1982)

    Article  MathSciNet  Google Scholar 

  15. R. Merris, Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197–198, 143–176 (1994)

    Article  MathSciNet  Google Scholar 

  16. I.N. Ponomarenko, Graph algebras and the graph isomorphism problem. Appl. Algebra Eng. Commun. 5(5), 277–286 (1994)

    Article  MathSciNet  Google Scholar 

  17. M.V. Ramana, E.R. Scheinmerman, D. Ullman, Fractional isomorphism of graphs. Discrete Math. 132(1–3), 247–265 (1994)

    Article  MathSciNet  Google Scholar 

  18. H.N. Salas, Greshgorin’s theorem for matrices of operators. Linear Algebra Appl. 291(1–3), 15–36 (1999)

    Article  MathSciNet  Google Scholar 

  19. H. Shang, F. Kang, C. Xu, G. Chen, S. Zhang, The SVE method for regular graph isomorphism identification. Circuits Syst. Signal Process. 34(11), 3671–3680 (2015)

    Article  MathSciNet  Google Scholar 

  20. L. Tian, C. Liu, J. Xie, A partition method for graph isomorphism. Phys. Proc. 25, 1761–1768 (2012)

    Article  Google Scholar 

Download references

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  1. Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, Iran

    Morteza Moradi

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  1. Morteza Moradi
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Correspondence to Morteza Moradi.

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Moradi, M. A Time-Based Solution for the Graph Isomorphism Problem. Circuits Syst Signal Process 39, 2695–2715 (2020). https://doi.org/10.1007/s00034-019-01279-8

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  • DOI: https://doi.org/10.1007/s00034-019-01279-8

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