Zeta Functions of Graphs: A Stroll through the Garden (Cambridge Studies in Advanced Mathematics Book 128) by Audrey Terras (PDF)

Description

Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced, showing you cannot hear the shape of a graph. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory, plus expander/Ramanujan graphs of interest in computer science. Created for beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Ihara zeta functionとAltin L-functionを可視化すると宇宙と異次元の位置関係を述べている。${d /over df}/int /int{1 /over (x /log x )^2}dx_m$の量子群がzeta functionの大域的微分多様体と加群したHiggs場の式は、このAltin professorが原子配列と同じと見抜いているのとほぼ同じで、結び目理論をこの原子と宇宙の形が同じ構造で表されているために微細構造定数としているのと同型の意味で重力と反重力であり、Higgs場とオイラーの定数を多様体積分した結果のzeta functionと同じ意味であるIhara zeta functionとAltin L-functionを加群したのがHiggs場の式で表現されていると思いました。

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