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letting diam(''G'') denote the [[diameter (graph theory)|diameter]] of ''G'', the following inequalities hold:<ref>{{harvnb|Chinn|Chvátalová|Dewdney|Gibbs|1982}}</ref>
:<math>\lceil (n-1)/\operatorname{diam}(G) \rceil \le \varphi(G) \le n - \operatorname{diam}(G),</math>
where <math>n</math> is the number of vertices in <math>G</math>.
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