R. Merris, K. R. Rebman, and W. Watkins, Permanental polynomials of graphs, Linear Algebra Appl. 38 (1981), 273-288. Math Reviews 83d: 15004.

This article initiated the study of graph G by means of per(xI-L(G)), the permanent of the characteristic matrix of the Laplacian matrix L(G). More recently, this invariant has begun to appear in the chemical literature. (See, e.g., Cash and Gutman [2004], below.)

Conjecture 3.4 was resolved affirmativly in

  • R. A. Brualdi and J. L. Goldwasser, Permanent of the Laplacian matrix of trees and bipartite graphs, Discrete Math. 48 (1984), 1-21;
  • A. Vrba, The permanent of the Laplacian matrix of a bipartite graph, Czech. Math. J. 36 (1986), 7-17;
  • and Merris's article 54.

    • Among the publications citing this paper are

  • G. N. de Oliveira, Linear & Multilinear Algebra 12 (1982), 125-138.
  • R. Merris, Czech J. Math. 32 (1982), 397-403.
  • G. N. de Oliveira and J. A. Dias da Silva, Linear Algebra Appl. 49 (1983), 191-219.
  • R. A. Brualdi & J. L. Goldwasser, Discrete Math. 48 (1984), 1-21.
  • I. Faria, Linear Algebra Appl. 64 (1985), 255-265.
  • M. Borowiecki, Publ. L'institut Math. 38 (1985), 31-33.
  • R. Grone, Linear Algebra Appl. 68 (1985), 252-254.
  • D. Cvetkovic and M. Doob, Linear & Multilinear Algebra 18 (1985), 153-181.
  • A. Vrba, Czech J. Math. 36 (1986), 7-17.
  • R. B. Bapat, Linear Algebra Appl. 74 (1986), 219-223.
  • J. L. Goldwasser, Discrete Math. 61 (1986), 197-212.
  • A. Krauter, Linear Algebra Appl. 93 (1987), 39-55.
  • R. Bhatia, L. Elsner, and G. Krause, Linear Algebra Appl. 142 (1990), 195-209.
  • V. Strok, Permanents: Theory and Appl. [Russian], Krasnoyarsk Politekhn. Inst., 1990, 73-78.
  • R. Grone, Linear Algebra Appl. 150 (1991), 167-178.
  • K. Balasubramanian, Theor. Chimica Acta 85 (1993), 379-390.
  • I. Faria, Linear Algebra Appl. 229 (1995), 15-35.
  • R. Horaud and H. Sossa, Pattern Recognition 28 (1995), 1855-1870.
  • J. Klerlein, S. Sportsman, and C. Wallis, Congressus Numerantium 118 (1996), 185-191.
  • O. Chan and T. K. Lam, SIAM J. Matrix Anal. Appl. 21 (1999), 129-144.
  • G. G. Cash, J. Chem. Inf. & Computer Sci. 40 (2000), 1203-1206.
  • I. Gutman and G. G. Cash, MATCH-Comm. Math. & Comput. Chem. 45 (2002), 55-70.
  • G. G. Cash, J. Chem. Inf. & Comp. Sci. 42 (2002), 1132-1135.
  • G. G. Cash, J. Chem. Inf. & Comp. Sci. 43 (2003), 1942-1946.
  • W. G. Yan and F. J. Zhang, J. Math. Chem. 35 (2004), 175-188.
  • G. G. Cash and I. Gutman, MATCH-Comm. Math. & Comput. Chem. 51 (2004), 129-136.

    • Also see:

  • H. Minc, Theory of permanents 1978-1981, Linear & Multilinear Algebra 12 (1983), 227-263.
  • D. Cvetkovic, M. Doob, I. Gutman, and A. Torgasev, Recent Results in the Theory of Graph Spectra, North-Holland, Amsterdam, 1988.