About some robustness and complexity properties of G-graphs networks

Given a finite group G and a set S ⊂ G, we consider the different cosets of each cyclic group ⟨s⟩ with s ∈ S. Then the G-graph Φ(G, S) associated with G and S can be defined as the intersection graph of all these cosets. These graphs were introduced in Bretto and Faisant (2005) as an alternative to Cayley graphs: they still have strong regular properties but a more flexible structure. We investigate here some of their robustness properties (connectivity and vertex/edge-transitivity) recognized as important issues in the domain of network design. In particular, we exhibit some cases where G-graphs are optimally connected, i.e. their edge and vertex-connectivity are both equal to the minimum degree. Our main result concerns the case of a G-graph associated with an abelian group and its canonical base S, which is shown to be optimally connected. We also provide a combinatorial characterization for this class as clique graphs of Cartesian products of complete graphs and we show that it can be recognized in polynomial time. These results motivate future researches in two main directions: revealing new classes of optimally connected G-graphs and investigating the complexity of their recognitio

Suono e Spettacolo. Athanasius Kircher, un percorso nelle Immagini sonore.

The Society of Jesus made great propaganda efforts throughout the seventeenth century and chose the images and the play as a privileged means to communicate and persuade. Athanasius Kircher, a key figure of the seventeenth century, he decided to dominate the wild nature of sound through Phonurgia Nova, which includes a gallery of powerful symbolic images for Baroque aesthetics. The essay, through the grant of the images from the Library of the Department of Mathematics "Guido Castelnuovo" Sapienza University of Rome, aims to understand, through the pictures offered by Kircher, the sound phenomenon and the spectacle that this produces. In Phonurgia Nova a process of dramatization sound effects takes place, often through machines and "visions" applied to the theatrical reality, as experimental and astonishing environment beloved in baroque. Kircher illustrates the sound through explanatory figures, so to dominate the sound through the eyes. Sound is seen, admired and represented: its spectacle not only takes place through the implementation of sound machines or the "wonders" applied to the theater, but even through images, creating create a sense of wonder in in the erudite person of the seventeenth century

Inverse booking problem: Inverse chromatic number problem in interval graphs

We consider inverse chromatic number problems in interval graphs having the following form: we are given an integer K and an interval graph G = (V,E), associated with n = |V| intervals I i = ]a i ,b i [ (1 ≤ i ≤ n), each having a specified length s(I i ) = b i - a i , a (preferred) starting time a i and a completion time b i . The intervals are to be newly positioned with the least possible discrepancies from the original positions in such a way that the related interval graph can be colorable with at most K colors. We propose a model involving this problem called inverse booking problem.We show that inverse booking problems are hard to approximate within O(n 1 - ε ), ε > 0 in the general case with no constraints on lengths of intervals, even though a ratio of n can be achieved by using a result of [13]. This result answers a question recently formulated in [12] about the approximation behavior of the unweighted case of single machine just-in-time scheduling problem with earliness and tardiness costs. Moreover, this result holds for some restrictive cases