 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-2/17h/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Take the infinite family

G_{n} = K_{n}, n \geq 2.

Each complete graph is connected and has graph diameter one. If

A_{n}

A_{n} = J - I

A_{n}^{2} = (n - 2) A_{n} + (n - 1) I .

Consequently

A_{n}^{2} - (n - 2) A_{n} - (n - 1) I = 0,

I, A_{n}, A_{n}^{2}

, which are precisely the powers through

d (G_{n}) + 1 = 2

, are linearly dependent for every

n \geq 2

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