### Prove the weakening of theorem

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##### Reference no: EM13926777

1. Show that radG ≤ diamG ≤ 2radG for every graph G.

Hint: Estimate the distances within G as seen from a central vertex.

2. Prove the weakening of Theorem 1.3.4 obtained by replacing average with minimum degree. Deduce that | G | ≥ n0(d/2, g) for every graph G as given in the theorem.

Hint: Count vertices as in the proof of Proposition 1.3.3. For the even case, start with two adjacent vertices.

3. Show that a connected graph of diameter k and minimum degree d has at least about kd/3 vertices but need not have substantially more.

Hint: Pick two vertices x, y of maximum distance, and show that many of the distance classes Di from x have to be large.

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