The power-law behavior is ubiquitous in a majority of real-world networks, and it was shown to have a strong effect on various combinatorial, structural, and dynamical properties of graphs. For example, it has been shown that in real-life power-law networks, both the matching number and the domination number are relatively smaller, compared with homogeneous graphs. In this paper, we study analytically several combinatorial problems for two power-law graphs with the same number of vertices, edges, and the same power exponent. For both graphs, we determine exactly or recursively their matching number, independence number, domination number, the number of maximum matchings, the number of maximum independent sets, and the number of minimum dominating sets. We show that power-law behavior itself cannot characterize the combinatorial properties of a heterogenous graph. Since the combinatorial properties studied here have found wide applications in different fields, such as structural controllability of complex networks, our work offers insight in the applications of these combinatorial problems in power-law graphs.

中文翻译：

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幂律图中的一些组合问题

幂律行为在大多数现实世界网络中无处不在，并且已证明对图形的各种组合，结构和动力学特性有很大的影响。例如，已经显示出，在现实生活中的幂律网络中，与均质图相比，匹配数和支配数都相对较小。在本文中，我们分析分析了两个具有相同顶点，边和相同幂指数的幂律图的几个组合问题。对于这两个图，我们精确地或递归地确定它们的匹配数，独立数，支配数，最大匹配数，最大独立集的数量和最小支配集的数量。我们表明，幂律行为本身无法表征异构图的组合特性。由于此处研究的组合性质已在不同领域中找到了广泛的应用，例如复杂网络的结构可控性，因此我们的工作为在幂律图中这些组合问题的应用提供了见识。