The F-triangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the F-triangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the " F = M Conjecture " of Armstrong which predicts a surprising relation between the F-triangle and the Möbius function of his m-divisible partition poset associated t...

Symmetries in a network connectivity regulate how the graph's functioning organizes into clustered states. Classical methods for tracing symmetry group of require very high computational costs, and therefore they are hard, or even impossible, execution large sized graphs. We here unveil that there is direct connection between elements eigen-vector centrality clusters network. This gives fresh f...

conclusions migraine and cluster headache have complex pathophysiologies. the exact mechanism causing pain in both conditions is incompletely understood and more research needs to be undertaken in this area. results there is evidence that the trigeminovascular system and neurogenic inflammation play important roles, together with certain areas of the brain, leading to these conditions being ter...

In clustering (also known as unsupervised learning and class discovery), the classes are unknown a priori and need to be identified from the unsupervised data. The cluster analysis is concerned about estimating the number of classes and assigning each observation to a certain class. In this article we discuss a method for clustering via the Laplacian matrix. Also, based on a similar argument, w...

Clustering, assortativity, and communities are key features of complex networks. We probe dependencies between these features and find that ensembles of networks with high clustering display both high assortativity by degree and prominent community structure, while ensembles with high assortativity show much less enhancement of the clustering or community structure. Further, clustering can ampl...