Vertex decomposability and regularity of very well-covered graphs

A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that $G$ is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the edge ideal of $G$, we show that the Castelnuovo-Mumford regularity of $R/I(G)$ is equal to the maximum number of pairwise 3-disjoint edges of $G$. This improves Kummini's result on unmixed bipartite graphs.Comment: 11 page

Analyzing the Composition of HDI in European Countries

Human Development Index (HDI) measures development in a country by combining indicators of life expectancy, education level and income. In 2013, 187 countries were included in this index, which aims to expand the coverage area as additional statistics become more available. HDI, which is published by UNDP, may be the most comprehensive indicator, but it is not fully compatible enough to measure the human development level in a global perspective. Human Development Index explicitly explains the development of a country as being more than an economic growth tool or material wealth. In this way, this index is distinguished from many other performance indicators. This article aims to analyze the proportion of the three indicators on 37 European countries