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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 17H / d / Solution

Take the infinite family

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

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

A_{n}

is its adjacency matrix, then

A_{n} = J - I

and direct multiplication gives the adjacency-matrix quadratic relation for a complete graph

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

Consequently

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

so

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

.

Solved by gpt-5.6-sol high.

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