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The bloom of complex network study, in particular, with respect to scale-freeones, is considerably triggering the research of scale-free graph itself.Therefore, a great number of interesting results have been reported in thepast, including bounds of diameter. In this paper, we focus mainly on a problemof how to analytically estimate the lower bound of diameter of scale-freegraph, i.e., how small scale-free graph can be. Unlike some pre-existingmethods for determining the lower bound of diameter, we make use of aconstructive manner in which one candidate model $\mathcal{G^*} (\mathcal{V^*},\mathcal{E^*})$ with ultra-small diameter can be generated. In addition, with arigorous proof, we certainly demonstrate that the diameter of graph$\mathcal{G^{*}}(\mathcal{V^{*}},\mathcal{E^{*}})$ must be the smallest incomparison with that of any scale-free graph. This should be regarded as thetight lower bound.

An extremal problem: How small scale-free graph can be