In this work we introduce a concept of complexity for undirected graphs in terms of the spectral analysis of the Laplacian operator defined by the incidence matrix of the graph. Precisely, we compute the norm of the vector of eigenvalues of both the graph and its complement and take their product. Doing so, we obtain a quantity that satisfies two basic properties that are the expected for a measure of complexity. First,complexity of fully connected and fully disconnected graphs vanish. Second, complexity of complementary graphs coincide. This notion of complexity allows us to distinguish different kinds of graphs by placing them in a "croissant-shaped" region of the plane link density - complexity, highlighting some features like connectivity,concentration, uniformity or regularity and existence of clique-like clusters. Indeed, considering graphs with a fixed number of nodes, by plotting the link density versus the complexity we find that graphs generated by different methods take place at different regions of the plane. We consider some generated graphs, in particular the Erd\"os-R\'enyi, the Watts-Strogatz and the Barab\'asi-Albert models. Also, we place some particular, let us say deterministic, to wit, lattices, stars, hyper-concentrated and cliques-containing graphs. It is worthy noticing that these deterministic classical models of graphs depict the boundary of the croissant-shaped region. Finally, as an application to graphs generated by real measurements, we consider the brain connectivity graphs from two epileptic patients obtained from magnetoencephalography (MEG) recording, both in a baseline period and in ictal periods .In this case, our definition of complexity could be used as a tool for discerning between states, by the analysis of differences at distinct frequencies of the MEG recording.

中文翻译：

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基于拉普拉斯算子谱分析的图复杂度测度

在这项工作中，我们根据由图的关联矩阵定义的拉普拉斯算子的谱分析，引入了无向图的复杂性概念。准确地说，我们计算图及其补集的特征值向量的范数并取其乘积。这样做，我们获得了满足两个基本属性的数量，这些属性是衡量复杂性的预期。首先，完全连接和完全断开的图的复杂性消失了。其次，互补图的复杂性重合。这种复杂性的概念允许我们通过将它们放置在平面链接密度的“羊角形”区域中来区分不同类型的图 - 复杂性，突出一些特征，如连接性、集中度、均匀性或规律性以及类群的存在。的确，考虑具有固定数量节点的图，通过绘制链接密度与复杂性的关系，我们发现由不同方法生成的图发生在平面的不同区域。我们考虑一些生成的图，特别是 Erd\"os-R\'enyi、Watts-Strogatz 和 Barab\'asi-Albert 模型。此外，我们放置了一些特定的，让我们说确定性的格子，星，超集中和包含集团的图。值得注意的是，这些确定性的经典图模型描绘了羊角面包形状区域的边界。最后，作为对实际测量生成的图的应用，我们考虑了大脑连接图来自在基线期和发作期从脑磁图 (MEG) 记录中获得的两名癫痫患者。在这种情况下，