, Volume 5, Issue 1, pp 33–39 | Cite as

Inverses of trees

  • C. D. Godsil


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.

AMS subject classification (1980)

05 C 25 05 C 50, 05 C 05, 06 A 10 


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  1. [1]
    M. Aigner,Combinatorial Theory, Springer, Berlin, 1979.zbMATHGoogle Scholar
  2. [2]
    B. Bollobás,Extremal Graph Theory, Academic Press, London, 1978.zbMATHGoogle 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.Google Scholar
  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.CrossRefGoogle Scholar
  6. [6]
    C. D. Godsil andB. D. McKay. A new graph product and its spectrum,Bull. Australian Math. Soc.,18 (1978), 21–28.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    B. Grünbaum,Convex Polytopes, Wiley, London. 1967.zbMATHGoogle Scholar
  8. [8]
    I. Gutman, Acyclic systems with extremal Hückelπ-electron energy,Theoret. Chim. Acta,45 (1977), 79–87.CrossRefGoogle 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.CrossRefGoogle Scholar
  10. [10]
    D. J. Klein, Treediagonal matrices and their inverses.Linear Algebra and its Applications.42 (1982), 109–117.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    L. Lovász,Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979.zbMATHGoogle Scholar
  12. [12]
    L. Lovász andJ. Pelikán, On the eigenvalues of trees,Periodica Math. Hung.,3 (1973). 175–182.zbMATHCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó 1985

Authors and Affiliations

  • C. D. Godsil
    • 1
  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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