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arXiv · 1911.09253

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

Fei Ma·Ping Wang·Bing Yao
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Abstract

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

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Fei Ma, Ping Wang, Bing Yao. 2019-11-21. An extremal problem: How small scale-free graph can be. https://arxiv.org/abs/1911.09253

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