We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. In particular, we study the Gaussian Estrada index-an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdős-Rï¿oenyi random graphs as well as for the Barabï¿oesi-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs. 2The spectrum of a network-the set of its eigenvalues-provides important information about the structural and dynamical properties of the corresponding system. Most of the functions used to study network spectra give more weight to the largest modular eigenvalues. Then, the information contained in the eigenvalues close to the centre of the spectra, i.e, those close to zero, has remained totally unexplored in the study of graph spectra. Here we study a Gaussian matrix function that gives more weights to the eigenvalues closest to the centre of the spectrum of a network. Using this function we extract important structural information hidden in the spectra of networks, such as emergence of complete bipartite subgraphs (bicliques) which appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. These bicliques are also ubiquitous in random networks generated by preferential attachment mechanisms, such as the Barabï¿oesi-Albert model. In this work we provide a series of analytical results that pave the way for further analysis and uses of this Gaussian matrix function to understand network structure and dynamics.3

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In this paper, we focus on enriched cl-premonoid-valued topological groups, and their so-called change-of-basis lattice. In so doing, we take L as an enriched cl-premonoid and present a category SL-NTopGrp, of strati¯ed enriched cl-premonoid-valued neighborhood topological groups. We produce some characterization theorems, and prove that every strati¯ed L-neighborhood topological group is uniformizable. Finally, we look at the enriched lattice-valued neighborhood topological group when the underlying basis is changed under certain functorial mechanism.

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