Incidence Matrix

  • Ravindra B. Bapat
Part of the Universitext book series (UTX)


This chapter is devoted to the study of the adjacency matrix of a graph. The eigenvalues of a graph are defined to be the eigenvalues of its adjacency matrix. We first obtain the eigenvalues of the cycle and the path explicitly. A combinatorial description of the determinant of the adjacency matrix is provided. Some basic bounds involving the extreme eigenvalues of the adjacency matrix are provided, with detailed proofs. The energy of a graph, which finds applications in mathematical chemistry, is introduced. It is shown that the energy of a graph cannot be an odd integer. An elegant result of Stanley on counting directed paths is proved. In the final section we discuss trees which have a nonsingular adjacency matrix, and identify the cases when the inverse of the adjacency matrix corresponds to a graph.

References and Further Reading

  1. [Bap02]
    Bapat, R.B., Pati, S.: Path matrices of a tree. J. Math. Sci. 1, 46–52 (2002). New Series (Delhi)MathSciNetMATHGoogle Scholar
  2. [BHK81]
    Bevis, J.H., Hall, F.J., Katz, I.J.: Integer generalized inverses of incidence matrices. Linear Algebra Appl. 39, 247–258 (1981)MathSciNetCrossRefMATHGoogle Scholar
  3. [Big93]
    Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1993)Google Scholar
  4. [GR01]
    Godsil, C., Royle, G.: Algebraic Graph Theory, Graduate Texts in Mathematics. Springer, New York (2001)CrossRefGoogle Scholar
  5. [GKS95]
    Grossman, J.W., Kulkarni, D., Schochetman, I.E.: On the minors of an incidence matrix and its Smith normal form. Linear Algebra Appl. 218, 213–224 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. [Ij65]
    Ijiri, Y.: On the generalized inverse of an incidence matrix. J. Soc. Ind. Appl. Math. 13(3), 827–836 (1965)MathSciNetCrossRefMATHGoogle Scholar
  7. [LP86]
    Lovász, L., Plummer, M.D.: Matching Theory, Annals of Discrete Mathematics. North-Holland, Amsterdam (1986)Google Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.Indian Statistical InstituteNew DelhiIndia

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