“Investigate graphs” names a field, not a manageable question. A bounded alternative asks how many spanning trees remain when one edge is removed from a cycle or a complete graph. It specifies objects, a quantity and a perturbation, while leaving room for examples, a conjecture and a proof.
Senior investigation can be original exposition, a careful extension or genuinely new research. These are different contributions. Not finding a result in a short search does not establish that it is unknown.
Predict
A cycle with n vertices has n spanning trees, one for each deleted edge. Predict the count after one edge is permanently removed. Then ask whether the same simple subtraction works for every graph. A small comparison can reveal which structural feature drives the formula.
Experiment
The altered cycle is a path, so it has one spanning tree. For a triangle, the count falls from 3 to 1. For K₄, symmetry gives each of its six edges the same participation count across 16 spanning trees. There are 16·3=48 tree-edge incidences, so each edge belongs to 8 trees; deleting one leaves 8.
Graph
Original count
After deleting one edge
Cycle C₄
4
1
Complete graph K₄
16
8
The K₄ calculation depends on the known count 16 and edge symmetry. Mark those dependencies instead of presenting the arithmetic as a self-contained proof of every input fact.
Understand
A literature matrix records bibliographic identity, exact claim, hypotheses, proof method, limitations and relevance. For this graph question, one source might provide the matrix-tree theorem, another a deletion-contraction identity and another an effective-resistance interpretation. Verify the statements in primary sources; a search snippet or an unattributed summary may omit essential conditions.
Compare assumptions before transferring results. A theorem for finite connected simple graphs may need modification for loops, parallel edges or disconnected inputs. A proof using positivity may fail for signed weights. Write the mismatch explicitly and either prove a repair, restrict the question or leave the extension unresolved.
The Mississippi College Math401 senior-seminar syllabus, whose page does not specify a year, emphasizes a mathematical paper, significant proof and presentation. This unit uses those forms as preparation; it does not reproduce that institution’s assessment or award its credit.
A feasible first result should be small enough to verify independently. Reproduce a theorem on a controlled example, prove a special case or locate a counterexample to an overbroad claim. A fallback can still be mathematically substantial: explaining why a conjecture fails and proving a corrected statement is better than hiding the failed attempt behind a vague survey.
Apply
Write a question brief with a defined object family, target claim, three verified sources, prerequisite tools and one feasible first result. Distinguish what you know, what you conjecture and what you have not checked. Record the search scope and avoid claiming novelty without an appropriately broad search and expert review.
Continue
Conjectures organize exploration, but counterexamples sharpen them. The next lesson treats failed claims as evidence and requires quantifiers precise enough for a single counterexample to have a clear logical role.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Turn “study networks” into a feasible mathematical investigation question.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Construct a source-comparison matrix for three relevant mathematical sources.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A theorem assumes compactness but your domain is unbounded. What must a literature critique say?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Why does not finding a result in three sources fail to establish novelty?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reproduce one small claim from a source before extending it.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Submit a question and literature brief for your investigation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
