7. Kirchhoff's Matrix-Tree Theorem Revisited: Counting Spanning Trees with the Quantum Relative Entropy

  1. Matthias Dehmer3,
  2. Abbe Mowshowitz4 and
  3. Frank Emmert-Streib4
  1. Vittorio Giovannetti1 and
  2. Simone Severini2

Published Online: 12 JUL 2013

DOI: 10.1002/9783527670468.ch07

Advances in Network Complexity

Advances in Network Complexity

How to Cite

Giovannetti, V. and Severini, S. (2013) Kirchhoff's Matrix-Tree Theorem Revisited: Counting Spanning Trees with the Quantum Relative Entropy, in Advances in Network Complexity (eds M. Dehmer, A. Mowshowitz and F. Emmert-Streib), Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany. doi: 10.1002/9783527670468.ch07

Editor Information

  1. 3

    UMIT, Institut für Bioinformatik und, Translationale Forschung, Eduard-Wallnöfer-Zentrum 1, 6060 Hall in Tyrol, Austria

  2. 4

    The City College of New York, Department of Computer Science, 138th Street at Convent Avenue, New York, NY 10031, USA

Author Information

  1. 1

    NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, Piazza dei Cavalieri 7, I-56126, Pisa, Italy

  2. 2

    University College London, Department of Computer Science and Department of Physics & Astronomy, Gower St., WC1E 6BT, London, UK

Publication History

  1. Published Online: 12 JUL 2013
  2. Published Print: 10 JUL 2013

ISBN Information

Print ISBN: 9783527332915

Online ISBN: 9783527670468



  • Kirchhoff 's matrix-tree theorem;
  • lower bound;
  • quantum relative entropy;
  • spanning trees;
  • upper bound


By reinterpreting the Kirchhoff's matrix-tree theorem in the context of quantum information theory, the authors provide an exact formula to count spanning trees based on the notion of quantum relative entropy. This function is the quantum mechanical analog of the relative entropy. The authors show that the number of spanning trees is proportional to the distinguishability/distance between a certain density matrix associated with the graph in context and the maximally mixed state, that is, the state with maximum von Neumann entropy, or, equivalently, maximum amount of classical uncertainty. The chapter contains the mathematical setup and the main result. It offers a discussion on the lower and upper bounds on t(G) by exploiting known facts about the quantum relative entropy. The authors show particular attention to a plausible operational meaning for the number of spanning trees, when considering class of quantum states.