Line graphs of complex unit gain graphs with least eigenvalue -2

Let T be the multiplicative group of complex units, and let L(φ) denote a line graph of a T-gain graph φ. Similarly to what happens in the context of signed graphs, the real number min Spec(A(L(φ)), that is, the smallest eigenvalue of the adjacency matrix of L(φ), is not less than -2. The structural conditions on φ ensuring that min Spec(A(L(φ)) = -2 are identified. When such conditions are fulfilled, bases of the -2-eigenspace are constructed with the aid of the star complement technique

On eigenspaces of some compound complex unit gain graphs

Let T be the multiplicative group of complex units, and let L (Φ) denote the Laplacian matrix of a nonempty T-gain graph Φ = (Γ, T, γ). The gain line graph L (Φ) and the gain subdivision graph S (Φ) are defined up to switching equivalence. We discuss how the eigenspaces determined by the adjacency eigenvalues of L (Φ) and S (Φ) are related with those of L (Φ)

On the multiplicity of α as an A_α (Γ)-eigenvalue of signed graphs with pendant vertices

A signed graph is a pair Γ = (G; ), where x = (V (G);E(G)) is a graph and E(G) -> {+1;−1} is the sign function on the edges of G. For any > [0; 1] we consider the matrix Aα(Γ) = αD(G) + (1 −α )A(Γ); where D(G) is the diagonal matrix of the vertex degrees of G, and A(Γ) is the adjacency matrix of Γ. Let mAα(Γ) be the multiplicity of α as an A(Γ)-eigenvalue, and let G have p(G) pendant vertices, q(G) quasi-pendant vertices, and no components isomorphic to K2. It is proved that mA(Γ)() = p(G) − q(G) whenever all internal vertices are quasi-pendant. If this is not the case, it turns out that mA(Γ)() = p(G) − q(G) +mN(Γ)(); where mN(Γ)() denotes the multiplicity of as an eigenvalue of the matrix N(Γ) obtained from A(Γ) taking the entries corresponding to the internal vertices which are not quasipendant. Such results allow to state a formula for the multiplicity of 1 as an eigenvalue of the Laplacian matrix L(Γ) = D(G) − A(Γ). Furthermore, it is detected a class G of signed graphs whose nullity – i.e. the multiplicity of 0 as an A(Γ)-eigenvalue – does not depend on the chosen signature. The class G contains, among others, all signed trees and all signed lollipop graphs. It also turns out that for signed graphs belonging to a subclass G ` G the multiplicity of 1 as Laplacian eigenvalue does not depend on the chosen signatures. Such subclass contains trees and circular caterpillars

Introduction to the Proceedings of WGTTG2021

We describe the Italy-South Africa Research Program 2018–2020, focusing on the mobility scheme “Algebraic Graph Theory and Complex Networks” which supported a series of scientific initiatives between the University of Cape Town (Cape Town, South Africa) and the University of Naples Federico II (Naples, Italy) during the years 2018–2021. We sketch the relevant steps of the collaboration, focusing on the creation of a network of researchers between Italy and South Africa in the fields of Graph Theory and Combinatorics. In this context it becomes more relevant the role of the “Workshop on Graphs, Topology and Topological Groups 2021” and of the corresponding proceedings

On the accuracy of interpolation based on single-layer artificial neural networks

In the present paper, we consider one-hidden layer ANNs with a feedforward architecture, also referred to as shallow or two-layer networks, so that the structure is determined by the number and types of neurons. The determination of the parameters that define the function, called training, is done via the resolution of the approximation problem, so by imposing the interpolation through a set of specific nodes. We present the case where the parameters are trained using a procedure that is referred to as Extreme Learning Machine (ELM) that leads to a linear interpolation problem. In such hypotheses, the existence of an ANN interpolating function is guaranteed. The focus is then on the accuracy of the interpolation outside of the given sampling interpolation nodes when they are the equispaced, the Chebychev, and the randomly selected ones. The study is motivated by the well-known bell-shaped Runge example, which makes it clear that the construction of a global interpolating polynomial is accurate only if trained on suitably chosen nodes, ad example the Chebychev ones. In order to evaluate the behavior when growing the number of interpolation nodes, we raise the number of neurons in our network and compare it with the interpolating polynomial. We test using Runge's function and other well-known examples with different regularities. As expected, the accuracy of the approximation with a global polynomial increases only if the Chebychev nodes are considered. Instead, the error for the ANN interpolating function always decays and in most cases we observe that the convergence follows what is observed in the polynomial case on Chebychev nodes, despite the set of nodes used for training

Line and subdivision graphs determined by T4-gain graphs

Let T4 = (±1, ±i) be the subgroup of fourth roots of unity inside T, the multiplicative group of complex units. For a T4-gain graph Φ = (Γ,T4, ϕ), we introduce gain functions on its line graph L(Γ) and on its subdivision graph S(Γ). The corresponding gain graphs L(Φ) and S(Φ) are defined up to switching equivalence and generalize the analogous constructions for signed graphs. We discuss some spectral properties of these graphs and in particular we establish the relationship between the Laplacian characteristic polynomial of a gain graph Φ, and the adjacency characteristic polynomials of L(Φ) and S(Φ). A suitably defined incidence matrix for T4-gain graphs plays an important role in this contex

Crashworthiness of a composite bladder fuel tank for a tilt rotor aircraft

The fulfilment of the crash is a demanding requirement for a Tiltrotor. Indeed, such a kind of aircraft, being a hybrid between an airplane and a helicopter, inherits the requirements mainly from helicopters (EASA CS 29) due to its hovering ability. In particular, the fuel storage system must be designed in such a manner that it is crash resistant, under prescribed airworthiness requirements, in order to avoid the fuel leakage during such an event, preventing fire and, thus, increasing the survival chances of the crew and the passengers. The present work deals with the evaluation of crashworthiness of the fuel storage system of a Tiltrotor (bladder tank), and, in particular, it aims at describing the adopted numerical approach and some specific results. Crash resistance requirements are considered from the earliest design stages, and for this reason they are mainly addressed from a numerical point of view and by simulations that treat both single components and small/medium size assemblies. The developed numerical models include all the main parts needed for simulating the structural behavior of the investigated wing section: the tank, the structural components of the wing, the fuel sub-systems (fuel lines, probes, etc.) and the fuel itself. During the crash event there are several parts inside the tanks that can come into contact with the tank structure; therefore, it is necessary to evaluate which of these parts can be a damage source for the tank itself and could generate fuel loss. The SPH approach has been adopted to discretise fuel and to estimate the interaction forces with respect to the tank structure. Experimental data were used to calibrate the fuel tank and foam material models and to define the acceleration time-history to be applied. Thanks to the optimized foam’s configuration, the amount of dissipated impact energy is remarkable, and the evaluation of tanks/fuel system stress distribution allows estimating any undesired failure due to a survivable crash event

Vitamin d deficiency induces chronic pain and microglial phenotypic changes in mice

The bioactive form of vitamin .D, 1,25‐dihydroxyvitamin D (1,25D3), exerts immunomodulatory actions resulting in neuroprotective effects potentially useful against neurodegenerative and autoimmune diseases. In fact, vitamin D deficiency status has been correlated with painful manifestations associated with different pathological conditions. In this study, we have investigated the effects of vitamin D deficiency on microglia cells, as they represent the main immune cells responsible for early defense at central nervous system (CNS), including chronic pain states. For this purpose, we have employed a model of low vitamin D intake during gestation to evaluate possible changes in primary microglia cells obtained from postnatal day(P)2‐ 3 pups. Afterwards, pain measurement and microglia morphological analysis in the spinal cord level and in brain regions involved in the integration of pain perception were performed in the parents subjected to vitamin D restriction. In cultured microglia, we detected a reactive—activated and proliferative—phenotype associated with intracellular reactive oxygen species (ROS) generation. Oxidative stress was closely correlated with the extent of DNA damage and increased β‐galactosidase (B‐gal) activity. Interestingly, the incubation with 25D3 or 1,25D3 or palmitoylethanolamide, an endogenous ligand of peroxisome proliferator‐activated‐receptor‐alpha (PPAR‐α), reduced most of these effects. Morphological analysis of ex‐vivo microglia obtained from vitamin‐D‐deficient adult mice revealed an increased number of activated microglia in the spinal cord, while in the brain microglia appeared in a dystrophic phenotype. Remarkably, activated (spinal) or dystrophic (brain) microglia were detected in a prominent manner in females. Our data indicate that vitamin D deficiency produces profound modifications in microglia, suggesting a possible role of these cells in the sensorial dysfunctions associated with hypovitaminosis D