Degree distribution in random planar graphs

We prove that for each $k\ge0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, so that the expected number of vertices of degree $k$ is asymptotically $d_k n$, and moreover that $\sum_k d_k =1$. The proof uses the tools developed by Gimenez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function $p(w)=\sum_k d_k w^k$. From this we can compute the $d_k$ to any degree of accuracy, and derive the asymptotic estimate $d_k \sim c\cdot k^{-1/2} q^k$ for large values of $k$, where $q \approx 0.67$ is a constant defined analytically

Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S_g has a unique 2-connected component of linear size with high probability

On the diameter of random planar graphs

International audienceWe show that the diameter $D(G_n)$ of a random (unembedded) labelled connected planar graph with $n$ vertices is asymptotically almost surely of order $n^{1/4}$, in the sense that there exists a constant $c>0$ such that $P(D(G_n) \in (n^{1/4-\epsilon} ,n^{1/4+\epsilon})) \geq 1-\exp (-n^{c\epsilon})$ for $\epsilon$ small enough and $n$ large enough $(n \geq n_0(\epsilon))$. We prove similar statements for rooted $2$-connected and $3$-connected embedded (maps) and unembedded planar graphs

Physiological parameters for Prognosis in Abdominal Sepsis (PIPAS) Study : a WSES observational study

BackgroundTiming and adequacy of peritoneal source control are the most important pillars in the management of patients with acute peritonitis. Therefore, early prognostic evaluation of acute peritonitis is paramount to assess the severity and establish a prompt and appropriate treatment. The objectives of this study were to identify clinical and laboratory predictors for in-hospital mortality in patients with acute peritonitis and to develop a warning score system, based on easily recognizable and assessable variables, globally accepted.MethodsThis worldwide multicentre observational study included 153 surgical departments across 56 countries over a 4-month study period between February 1, 2018, and May 31, 2018.ResultsA total of 3137 patients were included, with 1815 (57.9%) men and 1322 (42.1%) women, with a median age of 47years (interquartile range [IQR] 28-66). The overall in-hospital mortality rate was 8.9%, with a median length of stay of 6days (IQR 4-10). Using multivariable logistic regression, independent variables associated with in-hospital mortality were identified: age > 80years, malignancy, severe cardiovascular disease, severe chronic kidney disease, respiratory rate >= 22 breaths/min, systolic blood pressure 4mmol/l. These variables were used to create the PIPAS Severity Score, a bedside early warning score for patients with acute peritonitis. The overall mortality was 2.9% for patients who had scores of 0-1, 22.7% for those who had scores of 2-3, 46.8% for those who had scores of 4-5, and 86.7% for those who have scores of 7-8.ConclusionsThe simple PIPAS Severity Score can be used on a global level and can help clinicians to identify patients at high risk for treatment failure and mortality.Peer reviewe