Sage Reference Manual: Graph Theory - Mirrors

Sage Reference Manual: Graph Theory, Release 6.1.1
sage: P.characteristic_polynomial(laplacian=True)
x^10 - 30*x^9 + 390*x^8 - 2880*x^7 + 13305*x^6 -
39882*x^5 + 77640*x^4 - 94800*x^3 + 66000*x^2 - 20000*x
charpoly(var=’x’, laplacian=False)
Returns the characteristic polynomial of the adjacency matrix of the (di)graph.
Let G be a (simple) graph with adjacency matrix A. Let I be the identity matrix of dimensions the same
as A. The characteristic polynomial of G is defined as the determinant det(xI − A).
Note: characteristic_polynomial and charpoly are aliases and thus provide exactly the
same method.
INPUT:
•x – (default: ’x’) the variable of the characteristic polynomial.
•laplacian – (default: False) if True, use the Laplacian matrix.
See Also:
•kirchhoff_matrix()
•laplacian_matrix()
EXAMPLES:
sage: P = graphs.PetersenGraph()
sage: P.characteristic_polynomial()
x^10 - 15*x^8 + 75*x^6 - 24*x^5 - 165*x^4 + 120*x^3 + 120*x^2 - 160*x + 48
sage: P.charpoly()
x^10 - 15*x^8 + 75*x^6 - 24*x^5 - 165*x^4 + 120*x^3 + 120*x^2 - 160*x + 48
sage: P.characteristic_polynomial(laplacian=True)
x^10 - 30*x^9 + 390*x^8 - 2880*x^7 + 13305*x^6 -
39882*x^5 + 77640*x^4 - 94800*x^3 + 66000*x^2 - 20000*x
check_embedding_validity(*args, **kwds)
Deprecated: Use _check_embedding_validity() instead. See trac ticket #15551 for details.
check_pos_validity(*args, **kwds)
Deprecated: Use _check_pos_validity() instead. See trac ticket #15551 for details.
clear()
Empties the graph of vertices and edges and removes name, boundary, associated objects, and position
information.
EXAMPLES:
sage: G=graphs.CycleGraph(4); G.set_vertices({0:’vertex0’})
sage: G.order(); G.size()
4
4
sage: len(G._pos)
4
sage: G.name()
’Cycle graph’
sage: G.get_vertex(0)
’vertex0’
sage: H = G.copy(implementation=’c_graph’, sparse=True)
sage: H.clear()
1.1. Generic graphs 25