= Solution
Take the infinite family
$$
G_n=K_n,\qquad n\geq2.
$$
Each <complete graph> is connected and has <graph diameter> one. If $A_n$ is its <adjacency matrix of a graph>[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 $n\geq2$.
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