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Combinatorica

, Volume 38, Issue 5, pp 1251–1263 | Cite as

Inverses of Bipartite Graphs

  • Yujun Yang
  • Dong Ye
Original Paper

Abstract

Let G be a bipartite graph with adjacency matrix A. If G has a unique perfect matching, then A has an inverse A1 which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph. The inverses of bipartite graphs with unique perfect matchings have a strong connection to Möbius functions of posets. In this note, we characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-negative matrices, which settles an open problem of Godsil on inverses of bipartite graphs in [Godsil, Inverses of Trees, Combinatorica 5 (1985) 33–39].

Mathematics Subject Classification (2000)

05C22 05C50 06A07 

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References

  1. [1]
    M. Aigner: Combinatorial Theory, Springer, Berlin, 1979.CrossRefzbMATHGoogle Scholar
  2. [2]
    R. B. Bapat and E. Ghorbani: Inverses of triangular matrices and bipartite graphs, Linear Algebra Appl. 447 (2014), 68–73.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    D. Cvetković, I. Gutman and S. Simić: On self-pseudo-inverse graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. z. 602–633 (1978), 111–117.MathSciNetzbMATHGoogle Scholar
  4. [4]
    R. Donaghey and L. W. Shapiro: Motzkin numbers, J. Combin. Theory Ser. A 23 (1977), 291–301.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    C. D. Godsil: Inverses of trees, Combinatorica 5 (1985), 33–39.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    C. D. Godsil: Personal communication with D. Ye, 2015.Google Scholar
  7. [7]
    F. Harary: On the notion of balance of a signed graph, Michigan Math. J. 2 (1953), 143–146.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    F. Harary: The determinant of the adjacency matrix of a graph, SIAM Rev. 4 (1962), 202–210.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    F. Harary and H. Minc: Which nonnegative matrices are self-inverse? Math. Mag. 49 (1976), 91–92.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    D. J. Klein: Treediagonal matrices and their inverses, Linear Algebra Appl. 42 (1982), 109–117.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    L. LovÁsz: Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979.zbMATHGoogle Scholar
  12. [12]
    C. McLeman and E. McNicholas: Graph invertibility, Graphs Combin. 30 (2014), 977–1002.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    H. Minc: Nonnegative Matrices, Wiley, New York, 1988.zbMATHGoogle Scholar
  14. [14]
    S. K. Panda and S. Pati: On the inverese of a class of graphs with unique perfect matchings, Elect. J. Linear Algebra 29 (2015), 89–101.CrossRefzbMATHGoogle Scholar
  15. [15]
    R. Simion and D. Cao: Solution to a problem of C. D. Godsil regarding bipartite graphs with unique perfect matching, Combinatorica 9 (1989), 85–89.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    D. J. A. Welsh: Matroid Theory, Dover Publications, 2010.zbMATHGoogle Scholar
  17. [17]
    D. Ye, Y. Yang, B. Mandal and D. J. Klein: Graph invertibility and median eigenvalues, Linear Algebra Appl. 513 (2017), 304–323.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    T. Zaslavsky: Signed graphs, Discrete Appl. Math. 4 (1982), 47–74.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    T. Zaslavsky: A mathematical bibliography of signed and gain graphs and allied areas, Electron. J. Combin. 8 (1998), #DS8: 1–124.MathSciNetzbMATHGoogle Scholar
  20. [20]
    T. Zaslavsky: Signed graphs and geometry, arXiv:1303.2770 [math.CO].Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceYantai UniversityYantai, ShandongChina
  2. 2.Department of Mathematical Sciences and Center for Computational SciencesMiddle Tennessee State UniversityMurfreesboroUSA

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