def numberOfTriangles (G):
'''
Returns the number of triangles in $G$ by using the fact that the $(i,i)$
entry of the cube of the adjacency matrix of $G$ indicates the number of
closed walks of length $3$ that start and end at vertex $v_i$
(\textit {i.e.\} the number of triangles that $v_i$ participates in). To
get the total number of triangles in the whole graph, we must note that if
three vertices $v_1, v_2, v_3$ form a triangle, the trace counts this
six times. So, we correct for this by dividing by $6$.
'''
A = adjacencyMatrix (G)
return (A**3).trace() / 6
def eigenvalues (G):
'''
Returns the eigenvalues of the adjacency matrix of $G$.
'''
eigenvals = adjacencyMatrix(G).eigenvals()
# The following is necessary because the dictionary returned by
# \code{sympy.Matrix.eigenvals()} uses \code{sympy.Integer}s as
# keys rather than ordinary integers. Thus, we get the annoying
# situation where (for example), eigenvals() might return the
# dict \code{\{4: 1, -2: 2, 0: 3\}}, but the expression
# \code {4 in eigenvals} would fail.
return dict ( [ (int (k), eigenvals [k]) for k in eigenvals.keys() ] )
def is_triangleFree (G):
'''
Returns True if $G$ is triangle-free, otherwise False.
'''
# By Mantel's Theorem (West, p. 41), the maximum number of edges in
# an $n$-vertex triangle-free simple graph is the floor of $n^2 / 4$.
n = order (G)
if size (G) > floor (n**2 / 4.0):
return False
# Otherwise, recall that the $(i,j)$ entry of the $n$-th power of the
# adjacency matrix of a graph indicates the number of closed walks
# of length $n$ that begin at $v_i$ and end at $v_j$. (West, p. 455,
# Proposition 8.6.7.)
# A triangle is a closed walk of length $3$ starting and ending at
# the same vertex. Thus, we see that $G$ is triangle-free if and
# only if $\trace (A^3) = 0$, where $A$ is the adjacency matrix of $G$.
A = adjacencyMatrix (G)
return (A**3).trace() == 0
