R. Merris, R. Grone, and W. Watkins, Laplacian unimodular equivalence of graphs, Combinatorial and Graph-Theoretic Problems in Linear Algebra (R. Brualdi, S. Friedland, and V. Klee, eds.), IMA Volumes in Mathematics and Its Applications 50, Springer-Verlag, NY, 1993, pp 175-180. Math Reviews 94f: 05104. Research supported by NSA grant MDA 90-H-4024.

This article is concerned with the number of 1's in the Smith Normal Form of the Laplacian matrix of a graph (considered as an integer matrix). Unfortunately, the graph G2 in Figure 1 is incorrectly drawn. There should be 6 vertices and 10 edges. The perpendicular bisector of the base should be deleted. Further work along these lines can be found in

  • R. Merris, Linear Algebra Appl. 197/198 (1994), 143-176.
  • R. Merris, Linear Algebra Appl. 201 (1994), 57-60.

    • The article has been cited in

  • D. M. Cvetkovic, M. Doob, and H. Sachs, Spectra of Graphs, 3rd ed., Johann Ambrosius Barth, Heidelberg, 1995.
  • D. M. Cvetkovic, P. Rowlinson, and S. Simic, Eigenspaces of Graphs, Encyclopedia of Mathematics and Its Applications 66, Cambridge University Press, 1997.
  • M. Lien and W. Watkins, Linear Algebra Appl. 306 (2000), 123-130.
  • D. Lorenzini, Linear & Multilinear Algebra 47 (2000), 281-306.
  • H. Christianson and V. Reiner, Linear Algebra Appl. 349 (2002), 233-244.