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[Theorem1] Let $G = (V, E)$ be a undirected graph where $V$ is the set of vertices and $E$ the set of edges. If $m = \min_{v \in V}\lvert \delta(v) \rvert$ then $G$ has diameter $\Delta \leq 2 m + 1$.\r
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[Proof] Since $m$ is the minimum cardinality of a set of vertices in $G$ with neighbors, the desired result is an immediate consequence of Hall’s theorem.\r
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[Corollary] The diameter of $G$ is $\Delta \leq 2 m + 1$.