Ihara Zeta function, coefficients of Maclaurin series and Ramanujan graphs
نویسندگان
چکیده
منابع مشابه
Edge Reconstruction of the Ihara Zeta Function
We show that if a graph G has average degree d ≥ 4, then the Ihara zeta function of G is edge-reconstructible. We prove some general spectral properties of the Bass–Hashimoto edge adjancency operator T : it is symmetric on a Kreı̆n space and has a “large” semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if d > 4, one can reconstruct the number of n...
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Imagine you were going for a run downtown. You have a set distance you want to go, but you don’t like running the same path two days in a row. You don’t like stopping to turn around and take the same road since this breaks your stride, and just repeating some loop multiple times makes you bored. So, how many options do you have? This situation could be modeled with graph theory, where each inte...
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The infinite grid is the Cayley graph of Z × Z with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in its domain is finite. The grid zeta function is the first comp...
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We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur’s identity on the characteristic polynomial of a sum of matrices. The second evaluation of the ...
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In 2009, Cooper presented an infinite family of pairs of graphs which were conjectured to have the same Ihara zeta function. We give a proof of this result by using generating functions to establish a one-to-one correspondence between cycles of the same length without backtracking or tails in the graphs Cooper proposed. Our method is flexible enough that we are able to generalize Cooper’s graph...
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ژورنال
عنوان ژورنال: International Journal of Mathematics
سال: 2020
ISSN: 0129-167X,1793-6519
DOI: 10.1142/s0129167x20500822