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In this thesis, the Perron–Frobenius theorem which in its most general formstates that the spectral radius of a non-negative real square matrix is an eigenvaluewith a non-negative eigenvector, is proven. Related properties arederived, in particular the Collatz–Wielandt formula and a general form of anon-negative idempotent matrices. Furthermore, let Rn be the sub-semi-ringof Z≥0[Sn] generated by the Kazhdan–Lusztig basis. a description of R2-semimodules,R3-semi-modules and a classification of elementary R3-semi-modulesis given.

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This paper examines the Rado graph, the unique, countably infinite, universalgraph. Many of the central properties are covered in detail, and various constructionsare provided, using results from a variety of fields of mathematics. A variantof the Rado graph was initially constructed by Ackermann. The actual Rado graphwas studied later, by Erdős and Rényi, before Rado rediscovered it from a differentperspective. A multitude of other authors have since then contributed to the subject.

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We consider finite-dimensional complex semi-simple Lie algebras g. Any such Lie algebra has a Cartan subalgebra h, and its adjoint representation on g yields a root space decomposition of g, which in turn gives rise to a root system. These are in turn classified by the Dynkin diagrams. Conversely, for any root system, there is a corresponding semi-simple Lie algebra, and the complex semi-simple Lie algebras are therefore classified by the root system. Given a root system, Serre’s theorem states explicitly how to reconstruct corresponding semi-simple Lie algebra from this root system.