Week 7 — graphs & matrices

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Week map

Classify → Represent → Execute → Trap-check

• Recognize: Ask: What does matrix entry (i,j) usually mean?
• Procedure: Sum or count cells meeting rule (nonzero, or equals 1). Row i entries are out-edges from i in directed graph.
• Variations / traps: Watch for: Row vs column meaning swapped for directed edges.

Formula chain (compressed)

graph V,E → adjacency list/matrix → matrix[i][j] cell meaning.

1. Graph — G = (V, E) — vertices + edges
2. Adjacency list — neighbours of v — sparse graphs
3. Adjacency matrix — A[i][j] = 1 if edge — dense / quick lookup
4. Undirected — A symmetric — edge i–j = j–i
5. Degree — count incident edges — row/column sums

Deep study

Computational Thinking · Week 7 — Graphs and matrices

Week map

Graph notation

• Vertex (node) → entity → labeled $v_1, v_2, \ldots$ or named A, B, C.
• Edge → connection between two vertices → $(u, v)$.
• Undirected edge → $\{u, v\}$ — travel both ways.
• Directed edge → $(u \to v)$ — one-way arrow from $u$ to $v$.
• Degree of vertex → number of edges incident → in directed graph: in-degree and out-degree separately.

Path

• Path → sequence of vertices where consecutive pairs are edges.
• Simple path → no repeated vertices.

Adjacency matrix notation

• $M$ is $n \times n$ matrix.
• $M[i][j] = 1$ (or weight) if edge from $i$ to $j$ exists; else $0$.
• Undirected graph → matrix symmetric: $M[i][j] = M[j][i]$.

text
M = [0 1 0]
    [1 0 1]
    [0 1 0]

Matrix indexing

• $M[i][j]$ → row $i$, column $j$ → 0-based unless problem states 1-based.
• Row $i$ → all $j$ values — outgoing from $i$.
• Column $j$ → all $i$ values — incoming to $j$ (directed).

Pattern families

Easy — Read graph from diagram

Medium — Build or read adjacency matrix

Hard — Degree and path reasoning

Worked mini-examples

Traps

• 0-based vs 1-based vertex labels in matrix.
• Directed edge $i \to j$ does not imply $j \to i$.
• Counting self-loop $M[i][i]=1$ in degree twice wrongly in undirected mental model.
• Confusing number of vertices with matrix size $n$.
• Path vs edge — path can use multiple edges.

Diagnostic (try yourself)

1. Graph with vertices {P, Q, R} and edges P—Q, Q—R. What is degree of Q?
2. Write adjacency matrix (0/1) for single undirected edge between vertices 0 and 1, no other edges ($n=2$).
3. In directed graph, $M[2][5]=1$. What does this mean?
4. How many edges in undirected graph if adjacency matrix has 6 ones above the diagonal?
5. Vertex with row sum 0 in directed adjacency matrix — what can you conclude?

ChatGPT prep archive

• Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

• Graph: vertices (nodes) and edges (connections); may be directed or undirected.
• Adjacency: whether edge exists between two vertices.
• Matrix representation: rows and columns index vertices; cell encodes edge weight or count.
• Undirected: matrix often symmetric; directed: row → column direction.

Notation & vocabulary

Pattern families

Easy — Count edges from matrix

Medium — Directed vs undirected

Hard — Path of length 2

Drill these on the pattern atlas — filter to week 7.

Traps

• Row vs column meaning swapped for directed edges.
• Assuming symmetry without undirected statement.
• Diagonal entries (self-loops) policy ignored.
• Counting each undirected edge twice.

Retrieval prompts

1. What does matrix entry (i,j) usually mean?
2. How to tell undirected from matrix?
3. Where are out-edges of vertex i?

Practice loop

1. Read Deep study (if present) or core concepts once.
2. Recite the formula chain without looking.
3. Open one easy pattern on the interactive atlas for week 7.
4. Attempt without solutions; mark studied after an honest try.
5. Say one trap aloud before closing the tab.

Related

• Full weeks 1–8 prep guide
• Pattern atlas note
• Formula chain desk