Zero forcing sets and controllability of dynamical systems defined on graphs

In this paper, controllability of systems defined on graphs is discussed. We consider the problem of controllability of the network for a family of matrices carrying the structure of an underlying directed graph. A one-to-one correspondence between the set of leaders rendering the network controllable and zero forcing sets is established. To illustrate the proposed results, special cases including path, cycle, and complete graphs are discussed. Moreover, as shown for graphs with a tree structure, the proposed results of the present paper together with the existing results on the zero forcing sets lead to a minimal leader selection scheme in particular cases

Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs I: construction and numerical results

In a series of two papers we investigate the universal spectral statistics of chaotic quantum systems in the ten known symmetry classes of quantum mechanics. In this first paper we focus on the construction of appropriate ensembles of star graphs in the ten symmetry classes. A generalization of the Bohigas-Giannoni-Schmit conjecture is given that covers all these symmetry classes. The conjecture is supported by numerical results that demonstrate the fidelity of the spectral statistics of star graphs to the corresponding Gaussian random-matrix theories.Comment: 15 page

Compatibility of A4A_{4} Flavour Symmetric Minimal Extended Seesaw with (3+1)(3+1) Neutrino Data

Motivated by the recent resurrection of the evidence for an eV scale sterile neutrino from the MiniBooNE experiment, we revisit one of the most minimal seesaw model known as the minimal extended seesaw that gives rise to a $3+1$ light neutrino mass matrix. We consider the presence of $A_4$ flavour symmetry which plays a non-trivial role in generating the structure of the neutrino mass matrix. Considering a diagonal charged lepton mass matrix and generic vacuum alignments of $A_4$ triplet flavons, we classify the resulting mass matrices based on their textures. Keeping aside the disallowed texture zeros based on earlier studies of $3+1$ neutrino textures, we categorise the remaining ones based on texture zeros, $\mu-\tau$ symmetry in the $3\times3$ block and hybrid textures. After pointing out the origin of such $3+1$ neutrino textures to $A_4$ vacuum alignments, we use the latest $3+1$ neutrino oscillation data and numerically analyse the texture zeros and $\mu-\tau$ symmetric cases. We find that a few of them are allowed from each category predicting interesting correlations between neutrino parameters. We also find that all of these allowed cases prefer normal hierarchical pattern of light neutrino masses over inverted hierarchy.Comment: 46 pages, 8 figures, 21 table