Gonality of expander graphs
نویسندگان
چکیده
منابع مشابه
Expander Graphs, Gonality and Variation of Galois Representations
We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in oneparameter families of abelian varieties.
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This paper will introduce expander graphs. Kolmogorov and Barzdin’s proof on the three dimensional realization of networks will be discussed as one of the first examples of expander graphs. The last section will discuss error correcting code as an application of expander graphs to computer science.
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Let AG be the adjacency matrix of G. Let λ1 ≥ λ2 ≥ . . . ≥ λn be the eigenvalues of AG. Sometimes we will also be interested in the Laplacian matrix of G. This is defined to be LG = D−AG, where D is the diagonal matrix where Dvv equals the degree of the vertex v. For d-regular graphs, LG = dI −AG, and hence the eigenvalues of LG are d− λ1, d− λ2, . . . , d− λn. Lemma 1. • λ1 = d. • λ2 = λ3 = . ...
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Let G = (V,E) be an undirected d-regular graph, here, |V | = n, deg(u) = d for all u ∈ V . We will typically interpret the properties of expander graphs in an asymptotic sense. That is, there will be an infinite family of graphs G, with a growing number of vertices n. By “sparse”, we mean that the degree d of G should be very slowly growing as a function of n. When n goes to infinity (n → ∞), d...
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The gonality sequence (γr)r>1 of a finite graph/metric graph/algebraic curve comprises the minimal degrees γr of linear systems of rank r. For the complete graph Kd, we show that γr = kd − h if r < g = (d−1)(d−2) 2 , where k and h are the uniquely determined integers such that r = k(k+3) 2 − h with 1 6 k 6 d − 3 and 0 6 h 6 k. This shows that the graph Kd has the gonality sequence of a smooth p...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2018
ISSN: 0012-365X
DOI: 10.1016/j.disc.2018.06.012