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# Computational Thinking · Week 7 — Graphs and matrices Deep study for Quiz 2 week 7. Graphs model connections between vertices; matrices store grid or adjacency data in rows and columns.

Computational Thinking · Week 7 — Graphs and matrices

Deep study for Quiz 2 week 7. Graphs model connections between vertices; matrices store grid or adjacency data in rows and columns.

Week map

Vertex and edge → directed vs undirected → adjacency matrix → matrix indexing → paths and degrees.

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.

Mini-example: vertices {A, B, C}, edges A—B, B—C. A has degree 1, B has degree 2, C has degree 1.

Path

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

Path A→B→C exists above; A→C does not (no direct edge).

Adjacency matrix notation

For

n

vertices labeled

0..n-1

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

Mini-example: 3 vertices, edges 0—1 and 1—2 (undirected):


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

Row

i

lists neighbors of vertex

i

(out-neighbors if directed).

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

Count vertices and edges. List neighbors of one vertex. Identify directed vs undirected.

Medium — Build or read adjacency matrix

Fill matrix from edge list. Read

M[i][j]

for edge existence. Check symmetry for undirected.

Hard — Degree and path reasoning

Compute degree from matrix row sums. Determine if path exists of length 2 (via intermediate vertex). Spot isolated vertex (all zeros in row and column).

Worked mini-examples

Example 1 — Neighbors.

Triangle on {0,1,2} all connected. Each vertex degree 2.

Example 2 — Directed matrix.

Edge

0 \to 1

only:

M[0][1]=1

M[1][0]=0

. Not symmetric.

Example 3 — Row sum = out-degree.

Row [0, 1, 1, 0] sums to 2 — vertex has two outgoing edges (directed).

Example 4 — Path length 2.

M[0][1]=1

and

M[1][2]=1

implies path 0→1→2 exists (length 2).

Example 5 — Isolated vertex.

Row and column all zero — no edges incident.

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)

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