Algebraic geometry 01 Algebraic curves, algebraic manifolds by I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

By I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

"... To sum up, this publication is helping to benefit algebraic geometry very quickly, its concrete kind is pleasing for college students and divulges the great thing about mathematics." --Acta Scientiarum Mathematicarum

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Sample text

Conversely, suppose that Γ is a connected regular graph of valency k having three distinct eigenvalues k, R1 and R2 . Let A be the adjacency matrix of Γ with respect to a certain ordering of its vertices. Since A is symmetric, its minimal polynomial is (X − k)(X − R1 )(X − R2 ). Put B := (A−R1 I)(A−R2 I), where I is the v×v identity matrix. Since (A−kI)B = 0, every nonzero column of B is an eigenvector of A with eigenvalue k. Since Γ is a connected regular graph of valency k, the multiplicity of k as eigenvalue of A is equal to 1.

A finite point-line geometry S = (P, B, I) is called a t-design with parameters (v, k, λ), or shortly a t-(v, k, λ)-design, if the following properties are satisfied: • S contains precisely v points; • B = ∅ and every line of S is incident with precisely k points; • every t distinct points of S are incident with precisely λ lines; • if B1 and B2 are two distinct lines of S, then {x ∈ P | x I B1 } = {x ∈ P | x I B2 }. The lines of a t-design are usually called blocks. A t-(v, k, 1)-design is also called a Steiner system and is often denoted by S(t, k, v).

There are k possibilities for y and for given y, there are k − λ − 1 ≥ 0 possibilities for z. So, N = k(k − λ − 1). On the other hand, there are v − k − 1 ≥ 0 possibilities for z and for given z, there are μ possibilities for y. So, we also have that N = (v − k − 1)μ. An easy example of a strongly regular graph is the pentagon. This graph is strongly regular with parameters (v, k, λ, μ) = (5, 2, 0, 1). The disjoint union of r ≥ 2 complete graphs on m ≥ 2 vertices is a strongly regular graph with parameters (v, k, λ, μ) = (rm, m − 1, m − 2, 0).

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