Inverses of trees

Abstract

LetT be a tree with a perfect matching. It is known that in this case the adjacency matrixA ofT is invertible and thatA −1 is a (0, 1, −1)-matrix. We show that in factA −1 is diagonally similar to a (0, 1)-matrix, hence to the adjacency matrix of a graph. We use this to provide sharp bounds on the least positive eigenvalue ofA and some general information concerning the behaviour of this eigenvalue. Some open problems raised by this work and connections with Möbius inversion on partially ordered sets are also discussed.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    M. Aigner,Combinatorial Theory, Springer, Berlin, 1979.

    MATH  Google Scholar 

  2. [2]

    B. Bollobás,Extremal Graph Theory, Academic Press, London, 1978.

    MATH  Google Scholar 

  3. [3]

    D. M. Cvetković, M. Doob andH. Sachs,Spectra of Graphs. Academic Press, N. Y. 1980.

    Google Scholar 

  4. [4]

    D. Cvetković, I. Gutman andS. Simić, On self pseudo-inverse graphs.Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. fiz., No.602–633, (1978) 111–117.

  5. [5]

    D. Cvetković, I. Gutman andN. Trinajstić, Graph theory and molecular orbitals. VII. The role of resonance structures.J. Chemical Physics,61 (1974), 2700–2706.

    Article  Google Scholar 

  6. [6]

    C. D. Godsil andB. D. McKay. A new graph product and its spectrum,Bull. Australian Math. Soc.,18 (1978), 21–28.

    MATH  MathSciNet  Article  Google Scholar 

  7. [7]

    B. Grünbaum,Convex Polytopes, Wiley, London. 1967.

    MATH  Google Scholar 

  8. [8]

    I. Gutman, Acyclic systems with extremal Hückelπ-electron energy,Theoret. Chim. Acta,45 (1977), 79–87.

    Article  Google Scholar 

  9. [9]

    I. Gutman andD. Rouvray, An approximate topological formula for the HOMO-LUMO separation in alternant hydrocarbons,Chem. Phys. Letters,62 (1979), 384–388.

    Article  Google Scholar 

  10. [10]

    D. J. Klein, Treediagonal matrices and their inverses.Linear Algebra and its Applications.42 (1982), 109–117.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    L. Lovász,Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979.

    MATH  Google Scholar 

  12. [12]

    L. Lovász andJ. Pelikán, On the eigenvalues of trees,Periodica Math. Hung.,3 (1973). 175–182.

    MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Godsil, C.D. Inverses of trees. Combinatorica 5, 33–39 (1985). https://doi.org/10.1007/BF02579440

Download citation

AMS subject classification (1980)

  • 05 C 25
  • 05 C 50, 05 C 05, 06 A 10