(Not an answer; just a comment.)

It is a somewhat tangled graph. Here is a representation of $G(100)$:

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          ![SqGraph100][1]

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One can see $50+50=100=10^2$ at the lower-right corner, $69+52=121=11^2$ near the upper-right corner, $70+51=121=11^2$, left midlevel, etc.

This $G(100)$ graph has diameter $5$. But $G(1000)$ has diameter $4$.

(Not an answer; just a comment.)

It is a somewhat tangled graph. Here is a representation of $G(100)$:

---

          ![SqGraph100][1]

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One can see $50+71=121=11^2$ near the lower-right corner, $84+85=169=13^2$ near the top, $82+62=144=12^2$, near the bottom, etc.

This $G(100)$ graph has diameter $5$. But $G(1000)$ has diameter $4$.

(Not an answer; just a comment.)

It is a somewhat tangled graph. Here is a representation of $G(100)$:

---

          ![SqGraph100][1]

---

One can see $50+50=100=10^2$ at the lower-right corner, $69+52=111=11^2$ near the upper-right corner, $70+51=121=11^2$, left midlevel, etc.

This $G(100)$ graph has diameter $5$. But $G(1000)$ has diameter $4$.

(Not an answer; just a comment.)

It is a somewhat tangled graph. Here is a representation of $G(100)$:

---

          ![SqGraph100][1]

---

One can see $50+50=100=10^2$ at the lower-right corner, $69+52=121=11^2$ near the upper-right corner, $70+51=121=11^2$, left midlevel, etc.

This $G(100)$ graph has diameter $5$. But $G(1000)$ has diameter $4$.