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Computational and Theoretical Challenges for Computing the Minimum Rank of a Graph

Author

Listed:
  • Illya V. Hicks

    (Computational Applied Mathematics & Operations Research Department, Rice University, Houston, Texas 77005)

  • Boris Brimkov

    (Mathematics and Statistics Department, Slippery Rock University, Slippery Rock, Pennsylvania 16057)

  • Louis Deaett

    (Mathematics and Statistics Department Quninnipiac University, Hamden, Connecticut 06518)

  • Ruth Haas

    (Department of Mathematics, University of Hawaii at Monoa, Honolulu, Hawaii 96822)

  • Derek Mikesell

    (Computational Applied Mathematics and Operations Research Department, Rice University, Houston, Texas 77005)

  • David Roberson

    (Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs, Lyngby, Denmark; Centre for the Mathematics of Quantum Theory (QMATH), Department of Mathematical Sciences, University of Copenhagen, Copenhagen 2100, Denmark)

  • Logan Smith

    (Computational Applied Mathematics and Operations Research Department, Rice University, Houston, Texas 77005)

Abstract

The minimum rank of a graph G is the minimum of the ranks of all symmetric adjacency matrices of G. We present a new combinatorial bound for the minimum rank of an arbitrary graph G based on enumerating certain subsets of vertices of G satisfying matroid theoretic properties. We also present some computational and theoretical challenges associated with computing the minimum rank. This includes a conjecture that this bound on the minimum rank actually holds with equality for all graphs.

Suggested Citation

  • Illya V. Hicks & Boris Brimkov & Louis Deaett & Ruth Haas & Derek Mikesell & David Roberson & Logan Smith, 2022. "Computational and Theoretical Challenges for Computing the Minimum Rank of a Graph," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 2868-2872, November.
    • Handle: RePEc:inm:orijoc:v:34:y:2022:i:6:p:2868-2872
    DOI: 10.1287/ijoc.2022.1219
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