On The Isoperimetric Spectrum of Graphs and Its Approximations

In this paper we consider higher isoperimetric numbers of a (finite directed) graph. In this regard we focus on the $n$th mean isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set of $n$ disjoint subsets of the vertex set of the graph. We show that the second mean isoperimetric constant in this general setting, coincides with (the mean version of) the classical Cheeger constant of the graph, while for the rest of the spectrum we show that there is a fundamental difference between the $n$th isoperimetric constant and the number obtained by taking the minimum over all $n$-partitions. In this direction, we show that our definition is the correct one in the sense that it satisfies a Federer-Fleming-type theorem, and we also define and present examples for the concept of a supergeometric graph as a graph whose mean isoperimetric constants are attained on partitions at all levels. Moreover, considering the ${\bf NP}$-completeness of the isoperimetric problem on graphs, we address ourselves to the approximation problem where we prove general spectral inequalities that give rise to a general Cheeger-type inequality as well. On the other hand, we also consider some algorithmic aspects of the problem where we show connections to orthogonal representations of graphs and following J.~Malik and J.~Shi ($2000$) we study the close relationships to the well-known $k$-means algorithm and normalized cuts method

Pembrolizumab and other immune checkpoint inhibitors in locally advanced or metastatic Merkel Cell Carcinoma: safety and efficacy

International audienceIntroductionMerkel Cell Carcinoma (MCC) is a rare aggressive skin cancer, mostly affecting elderly patients. Until recently, patients with advanced disease were treated with cytotoxic chemotherapies despite rapid chemoresistance and high toxicity. As with other cancers, immune checkpoint inhibitors (CPI), including pembrolizumab, allow durable responses with a manageable safety profile in these patients.Areas coveredThis review describes the rationale for using PD-1/PD-L1 inhibitors in MCC, as well as efficacy and safety results from the three open-label trials investigating pembrolizumab or other PD-1/PD-L1 inhibitors in patients with advanced MCC. Real-life experience and predictive pre-treatment biomarkers are discussed to assess which patients are likely to be candidates for such strategies. Ongoing fields of research include the use of CPI in the adjuvant or neoadjuvant setting and combined strategies in refractory patients.Expert Opinion: CPI have become the standard of care for frontline treatment in patients with advanced MCC. Earlier introduction of CPI in the disease course, including neo-adjuvant and adjuvant settings, is likely to improve the outcomes further. Given the rarity of this cancer, we still need to harmonize efforts in order to conduct large-scale trials and efficiently identify best optimal care