NAMSSN UI Chapter

1. Graphs & combinatorics

The graph tools in Sage are useful for discrete mathematics, algorithm analysis, and combinatorial questions. You can build graphs, inspect their degrees, and test small conjectures quickly.

# A simple cycle graph on 6 vertices
G = graphs.CycleGraph(6)

print("Vertices:", G.vertices())
print("Edges:", G.edges())
print("Degree sequence:", G.degree_sequence())
print("Is the graph connected?", G.is_connected())

# In a notebook, you can draw it:
# G.plot().show()

# Search over small cycle graphs
bad = []
for n in range(3, 11):
    H = graphs.CycleGraph(n)
    # Example property: check if the graph has a triangle
    if H.clique_number() < 3:
        bad.append(n)

print("n for which C_n has no triangle:", bad)

2. Linear algebra, determinants & eigenvalues

This section is designed to match the way a linear algebra course at UI might use determinants and eigenvalues, but with Sage doing the heavy arithmetic. It is also a natural place for determinant experiments, which supervisors often enjoy seeing.

# A simple upper triangular matrix
A = matrix([[1, 2, 3],
            [0, 1, 4],
            [0, 0, 1]])

print("A =")
print(A)
print("det(A) =", A.det())
print("eigenvalues:", A.eigenvalues())
print("Characteristic polynomial:", A.charpoly())

# Random integer matrices and determinants
def random_det_example(n, bound=3, samples=5):
    for k in range(samples):
        A = random_matrix(ZZ, n, n, x=-bound, y=bound+1)
        print("\\nSample", k+1)
        print(A)
        print("det(A) =", A.det())

# Start with 3x3
random_det_example(3)

3. Pattern search and counterexamples

Sage is especially valuable when you suspect a pattern but do not know whether it is always true. You can search for supporting examples and also aggressively hunt for counterexamples.

# Example: explore n^2 + 1 for primality
bad = []
for n in range(2, 500):
    if not is_prime(n^2 + 1):
        bad.append(n)

print("First 20 n such that n^2 + 1 is composite:", bad[:20])

# Now check a matrix power behaviour
A = matrix([[0, 1],
            [1, 1]])

powers = [A^k for k in range(1, 8)]
for k, M in enumerate(powers, start=1):
    print("\\nA^%d =" % k)
    print(M)

4. From notebook to write-up

A Sage notebook is not the final product. The final product is usually a short report, a project memo, or an exam solution. This section suggests a simple structure for turning experiments into something you can submit or present.

1. State the question clearly. One or two sentences in plain language.
2. Describe the experiment. What family of objects did you test? What did Sage compute?
3. Show a few key outputs. Not all of them – just enough to show the pattern.
4. State the conjecture or theorem. Now that the pattern is clear, write the general statement.
5. Sketch the argument. Try to prove it fully, or at least explain why it is reasonable.