Eigenvectors of graph Laplacians: a landscape

We review the properties of eigenvectors for the graph Laplacian matrix, aiming at predicting a specific eigenvalue/vector from the geometry of the graph. After considering classical graphs for which the spectrum is known, we focus on eigenvectors that have zero components and extend the pioneering results of Merris (1998) on graph transformations that preserve a given eigenvalue $\lambda$ or shift it in a simple way. These transformations enable us to obtain eigenvalues/vectors combinatorially instead of numerically; in particular we show that graphs having eigenvalues $\lambda= 1,2,\dots,6$ up to six vertices can be obtained from a short list of graphs. For the converse problem of a $\lambda$ subgraph $G$ of a $\lambda$ graph $G"$, we prove results and conjecture that $G$ and $G"$ are connected by two of the simple transformations described above

Epidemic model on a network: analysis and applications to COVID-19

We analyze an epidemic model on a network consisting of susceptible-infected-recovered equations at the nodes coupled by diffusion using a graph Laplacian. We introduce an epidemic criterion and examine different vaccination/containment strategies: we prove that it is most effective to vaccinate a node of highest degree. The model is also useful to evaluate deconfinement scenarios and prevent a so-called second wave. The model has few parameters enabling fitting to the data and the essential ingredient of importation of infected; these features are particularly important for the current COVID-19 epidemic

Pumping Iron in the Preoperative Period: Is It Beneficial in Reducing Blood Transfusions?

Carla Patel pictured.https://openworks.mdanderson.org/aprn-week-23/1002/thumbnail.jp

Изучение структуры цинктитановых боросиликатных стекол по данным рассеяния рентгеновских лучей под малыми углами

В статті вивчено мікронеоднорідну будову цинктитанових боросилікатних стекол та процесів фазового розділу в них за даним розсіяння нейтронів під малими кутами. Зроблено висновок про характер розподілення часток, що виділяються за розмірами, які змінюються в дослідних стеклах в залежності від вмісту в них TiO₂ та ZnO. Встановлено вплив наявності мікронеоднорідностей після варки на характер їх фазовогорозподілення.In paper the micronon-uniform structure zinc-titanium borosilicate glass and processes of phase separation in them according to diffusing under vanishing angles of neutrons is investigated. It is drawn a In paper the micronon-uniform structure zinc-titanium borosilicate glass and processes of phase separation in them according to diffusing under vanishing angles of neutrons is investigated. It is drawn a leading-out on distribution of depositing corpuscles character on sizes which changes in studied glasses depending on the contents in them TiO₂ and ZnO. Effect of presence micro micronon-uniforms after melting on character of their phase separationis established