Using the ExportData utility (with -m option, see Appendix A, Installation and Usage) and GAP™ one can compute the expanded form of the characteristic equation, λI - A 0 , where A is the adjacency matrix of a graph. Another interesting thing is the incidence matrix. Call it B, then B B 2I is the adjacency matrix of the line graph. And the Laplacian of a graph is worth thinking about as well (see [God00]).

Actually while being first of all primarily interested in the faces instead of the vertices the duals of our graphs are more important to us. Boundaries thereby mean that holes are not regarded as valid but get ignored.

Example: Let's again consider a tetrahedron:

gap>A:=[
	[0, 1, 1, 1], 
	[1, 0, 1, 1], 
	[1, 1, 0, 1], 
	[1, 1, 1, 0]
];;
gap>Collected(Factors(CharacteristicPolynomial(A)));
[ [ -3+x_1, 1 ], [ 1+x_1, 3 ] ]

See the section called “Galois Groups” for a subsequent investigation of those polynomials. For the cartographic version of L2(7) (see the section called “L2(7)”) we have

gap>A:=[
	[0, 2, 0, 0, 0], 
	[2, 1, 2, 1, 1], 
	[0, 2, 0, 1, 1],
	[0, 1, 1, 0, 0], 
	[0, 1, 1, 0, 0]
]
gap>B:=[
	[2, 1, 0, 0], 
	[1, 0, 1, 1], 
	[0, 1, 0, 0], 
	[0, 1, 0, 0]
]

, so you realize the matrices are symmetric and have integral values. Btw. there are also some collapsed adjacency matrices to be discussed.