THE STRUCTURE OF GRAPHS ON n VERTICES WITH THE DEGREE SUM OF ANY TWO NONADJACENT VERTICES EQUAL TO n-2

Let G be an undirected simple graph on n vertices and sigma2(G)=n-2 (degree sum of any two non-adjacent vertices in G is equal to n-2) and alpha(G) be the cardinality of an maximum independent set of G. In G, a vertex of degree (n-1) is called total vertex. We show that, for n>=3 is an odd number then alpha(G)=2 and G is a disconnected graph; for n>=4 is an even number then 2=<alpha(G)<=(n+2)/2, where if  alpha(G)=2 then G is a disconnected graph, otherwise  G is a connected graph, G contains  k total vertices and n-k vertices of degree delta=(n-2)/2, where 0<=k<=(n-2)/2. In particular, when k=0 then G is an (n-2)/2-Regular graph

Authoritarianism and academic freedom in neoliberal Turkey

This article examines the relationship between academic freedom and authoritarianism in Turkey. While not a new problem in the Turkish context, academic freedom has come particularly under attack following the attempted military coup on 15 July 2016, as well as with the Turkish intervention in the Syrian conflict. This paper is focused on scholars and academics currently working in Turkish universities. The paper explores the following questions: (1) how do these scholars define academic freedom in Turkey; (2) what is the relationship between universities and the Turkish society; (3) what are the changes that higher education is facing following the 2016 coup d'etat, in particular, in terms of pressures and barriers to academic production; (4) how do attacks affect scholars' possibilities to create, lecture, and resist government's policies? Drawing on Gramsci's theory of intellectuals and his notion of hegemony, as well as Foucault's theory of power and/as knowledge, we explore the relationship between authority and knowledge. We argue that the government's aggressive politics against Turkish scholars is a result of the failure to consolidate its power and hegemony through knowledge, and to establish an intellectual base in a Gramscian fashion

CONDITIONS FOR GRAPHS ON n VERTICES WITH THE SUM OF DEGREES OF ANY TWO NONADJACENT VERTICES EQUAL TO n-2 TO BE A HAMILTONIAN GRAPH

Let G be an undirected simple graph on  $n \geq 3$ vertices with the degree sum of any two nonadjacent vertices in G equal to $n - 2$.  We determine the condition for G to be a Hamiltonian graph

Spherically-symmetric solutions in general relativity using a tetrad-based approach

We present a tetrad-based method for solving the Einstein field equations for spherically-symmetric systems and compare it with the widely-used Lemaître– Tolman–Bondi (LTB) model. In particular, we focus on the issues of gauge ambiguity and the use of comoving versus ‘physical’ coordinate systems. We also clarify the correspondences between the two approaches, and illustrate their differences by applying them to the classic examples of the Schwarzschild and Friedmann–Lemaître– Robertson–Walker spacetimes. We demonstrate that the tetrad-based method does not suffer from the gauge freedoms inherent to the LTB model, naturally accommodates non-uniform pressure and has a more transparent physical interpretation. We further apply our tetrad-based method to a generalised form of ‘Swiss cheese’ model, which consists of an interior spherical region surrounded by a spherical shell of vacuum that is embedded in an exterior background universe. In general, we allow the fluid in the interior and exterior regions to support pressure, and do not demand that the interior region be compensated. We pay particular attention to the form of the solution in the intervening vacuum region and illustrate the validity of Birkhoff’s theorem at both the metric and tetrad level. We then reconsider critically the original theoretical arguments underlying the so-called Rh = ct cosmological model, which has recently received considerable attention. These considerations in turn illustrate the interesting behaviour of a number of ‘horizons’ in general cosmological models