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Register a new userAn extremal problem: How small scale-free graph can be
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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.
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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 cosmic censorship exists in our Universe, an over-spinning body without releasing its angular momentum cannot collapse to spacetime singularities. A simple kinematical estimate of two particles approaching each other supports this expectation and suggests the existence of a minimum size of an over-spinning body. But this does not imply that the geometry near the naked singularity cannot appear. By analyzing initial data, i.e., a snapshot of a spinning body, we see that an over-spinning body may produce a geometry close to the Kerr naked singularity around itself at least as a transient configuration.
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 $1$-crossing and pairwise non-$k$-crossing vectors in $mathbb{Z}^w$? We state a conjecture that the answer is $k^{w-1}$. We prove the conjecture for $wleq 3$ and provide weaker upper bounds for $wgeq 4$. Also, for all $k$ and $w$, we construct several quite different examples of families of desired size $k^{w-1}$. This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set.
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 the density of $F$ in $G$, after a suitable normalization, is approximately at least the density of $F$ in an $epsilon$-regular approximation of $G$. In recent work, motivated by applications in extremal and additive combinatorics, we showed that $C_5$ has this property. Here we construct a family of graphs with the property.
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 especially in complex systems. Other approaches use entanglement witnesses, these methods tend to be less demanding but lack reliability. Here, we demonstrate that ANN -- artificial neural networks provide a balance between both approaches. In this paper, we make a comparison of ANN performance against witness-based methods for random general 2-qubit quantum states without any prior information on the states. Furthermore, we apply our approach to real experimental data set.
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 $G$. Such a question clearly needs restrictions on the colorings to be meaningful. For edge-colorings using $n-1$ colors and without rainbow cycles, known in the literature as JL-colorings, there turns out to be a particularly nice way of counting the rainbow spanning trees and we solve this problem completely for JL-colored complete graphs $K_n$ and complete bipartite graphs $K_{n,m}$. In both cases, we find tight upper and lower bounds; the lower bound for $K_n$, in particular, proves to have an unexpectedly chaotic and interesting behavior. We further investigate this question for JL-colorings of general graphs and prove several results including characterizing graphs which have JL-colorings achieving the lowest possible number of rainbow spanning trees. We establish other results for general $n-1$ colorings, including providing an analogue of Kirchoffs matrix tree theorem which yields a way of counting rainbow spanning trees in a general graph $G$.
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