The distribution of eigenvalues of graphs

AbstractWe show that every limit point of the kth largest eigenvalues of graphs is a limit point of the (k + 1)th largest eigenvalues, and we find out the smallest limit point of the kth largest eigenvalues and an upper bound of the limit points of the kth smallest eigenvalues. For k ≥ 4, we prove that there exists a gap beyond the smallest limit point in which no point is the limit point of the kth largest eigenvalues. For the third largest eigenvalues of a graph G with at least three vertices, we obtain that (1) λ3(G) < −1 iff G ≅ P3; (2) λ3(G) = −1 iff Gc is isomorphic to a complete bipartite graph plus isolated vertices: (3) there exist no graphs such that −1 < λ3(G) < (1 − √5)2. Consequently, if Gc is not a complete bipartite graph plus isolated vertices, λ3(G) ≥ λ3(D∗n), where D∗n is the complement of the double star S(1, n − 3)

Topics in node packing and coloring

Ph.D.George L. Nemhause

Public policy objectives shaping tourism : from China to South Africa.

MM 2005ABSTRACT The relationship between South Africa and China has grown in all areas including tourism since 1998. China is considered to be one of its core source tourism markets. However, a recent decline in arrivals from China means there is a need to understand the nature of South Africa as a tourist destination for Chinese. The aim of this study was to explore successes and challenges of the current South African tourism situation for the purpose of promoting South Africa in China. We conducted non-structured and semi-structured interviews with people involved in the industry; colleted data from South Africa Tourism surveys and reports; and then interpreted the results, using content analysis. We found five successful factors and seven challenges. The research identified crime and prices as areas of concern, about which we made some recommendations

The spectral radius of a planar graph

AbstractA decomposition result for planar graphs is used to prove that the spectral radius of a planar graph on n vertices is less than 4 + √ 3 (n − 3). Moreover, the spectral radius of an outerplanar graph on n vertices is less than 1 + √2 + √2 + √n − 5

A Novel Dissolution and Synchronous Extraction of Rare Earth Elements from Bastnaesite by a Functionalized Ionic Liquid [Hbet][Tf2N]

Rare earth elements (REEs) are widely used in high-tech industries and are important basic raw materials. Bastnaesite is one of the most important minerals used in the rare earth extraction industry, and the efficient development of it is the key guarantee for the safe supply of rare earth raw materials. In this study, a novel method for dissolving bastnaesite using a carboxyl-functionalized ionic liquid ([Hbet][Tf2N]) is proposed. This innovative method provides a collaborative model with the dissolution and synchronous extraction of rare earth elements during the heating and cooling of the [Hbet][Tf2N]&ndash;H2O system. In the heating process, rare earth elements can be dissolved in a weakly acidic environment of ionic liquid without the trouble of HF escaping, and the leaching efficiencies of rare earth elements are above 95%. During the cooling of the leaching system, the rare earth ions in the dissolved state are extracted into the ionic liquid phase due to the two-phase stratification of [Hbet][Tf2N] and aqueous solution. It has been proved that rare earth ions recovery and ionic liquid regeneration can be achieved by back extraction using oxalic acid for the REEs-loaded ionic liquid

A Novel Dissolution and Synchronous Extraction of Rare Earth Elements from Bastnaesite by a Functionalized Ionic Liquid [Hbet][Tf₂N]

Rare earth elements (REEs) are widely used in high-tech industries and are important basic raw materials. Bastnaesite is one of the most important minerals used in the rare earth extraction industry, and the efficient development of it is the key guarantee for the safe supply of rare earth raw materials. In this study, a novel method for dissolving bastnaesite using a carboxyl-functionalized ionic liquid ([Hbet][Tf2N]) is proposed. This innovative method provides a collaborative model with the dissolution and synchronous extraction of rare earth elements during the heating and cooling of the [Hbet][Tf2N]–H2O system. In the heating process, rare earth elements can be dissolved in a weakly acidic environment of ionic liquid without the trouble of HF escaping, and the leaching efficiencies of rare earth elements are above 95%. During the cooling of the leaching system, the rare earth ions in the dissolved state are extracted into the ionic liquid phase due to the two-phase stratification of [Hbet][Tf2N] and aqueous solution. It has been proved that rare earth ions recovery and ionic liquid regeneration can be achieved by back extraction using oxalic acid for the REEs-loaded ionic liquid