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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.
The angular momentum of the Kerr singularity should not be larger than a threshold value so that it is enclosed by an event horizon: The Kerr singularity with the angular momentum exceeding the threshold value is naked. This fact suggests that if the
For positive integers $w$ and $k$, two vectors $A$ and $B$ from $mathbb{Z}^w$ are called $k$-crossing if there are two coordinates $i$ and $j$ such that $A[i]-B[i]geq k$ and $B[j]-A[j]geq k$. What is the maximum size of a family of pairwise $1$-cross
For which graphs $F$ is there a sparse $F$-counting lemma in $C_4$-free graphs? We are interested in identifying graphs $F$ with the property that, roughly speaking, if $G$ is an $n$-vertex $C_4$-free graph with on the order of $n^{3/2}$ edges, then
Detection of entangled states is essential in both fundamental and applied quantum physics. However, this task proves to be challenging especially for general quantum states. One can execute full state tomography but this method is time demanding esp
A spanning tree of an edge-colored graph is rainbow provided that each of its edges receives a distinct color. In this paper we consider the natural extremal problem of maximizing and minimizing the number of rainbow spanning trees in a graph $G$. Su