Network bipartivity and the transportation efficiency of European passenger airlines

The analysis of the structural organization of the interaction network of a complex system is central to understand its functioning. Here, we focus on the analysis of the bipartivity of graphs. We first introduce a mathematical approach to quantify bipartivity and show its implementation in general and random graphs. Then, we tackle the analysis of the transportation networks of European airlines from the point of view of their bipartivity and observe significant differences between traditional and low cost carriers. Bipartivity shows also that alliances and major mergers of traditional airlines provide a way to reduce bipartivity which, in its turn, is closely related to an increase of the transportation efficiency

Bipartizing fullerenes

A fullerene graph is a cubic bridgeless planar graph with twelve 5-faces such that all other faces are 6-faces. We show that any fullerene graph on n vertices can be bipartized by removing O(sqrt{n}) edges. This bound is asymptotically optimal.Comment: 14 pages, 4 figure

Noncoplanar and chiral spin states on the way towards N\'eel ordering in fullerene Heisenberg models

Using high-accuracy variational Monte Carlo based on group-convolutional neural networks (GCNNs), we obtain the symmetry-resolved low-energy spectrum of the spin-1/2 Heisenberg model on several highly symmetric fullerene geometries, including the famous C$_{60}$ buckminsterfullerene. We argue that as the degree of frustration is lowered in large fullerenes, they display characteristic features of incipient magnetic ordering: correlation functions show high-intensity Bragg peaks consistent with N\'eel-like ordering, while the low-energy spectrum is organised into a tower of states. Competition with frustration, however, turns the simple N\'eel order into a noncoplanar one. Remarkably, we find and predict chiral incipient ordering in a large number of fullerene structures.Comment: 14+3 pages, 14 figures, 3+4 table