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Zeta Functions of Finite Graphs and Representations of $p$-Adic Groups
Abstract
§ 0. Introduction § 1. Graphs and multigraphs § 2. Zeta functions of finite graphs § 3. Spectrum of a finite multigraph § 4. Harmonic functions and the Hodge decomposition § 5. Representations of C[Ti, T2]; a proof of (3.14) § 6. Representations of p-adic groups § 7. Special values of Zx(u) § 8. Miscellaneous results § 9. Computations of Zx(u) for well known families of X § 10.Examples: list of graphs for n<6, m<1.§ 11. References § 0. Introduction 0-1.In this paper we shall be concerned with the two different subjects, which have been developed separately.One is a combinatorial problem in algebraic graph theory, and the other is an arithmetic of discrete subgroups of p-adic groups and their representations.Suppose that Xis a finite (multi)graph, which is not a tree.We always assume that X is non-oriented.A closed path C in X is called reduced, if C and C 2 = C.C have no backtracking.Then obviously the set ~r•d(X) of reduced closed paths of length / is finite, and #(~['a(X)) -oo (l-oo) if Xis not homotopic to a circuit, i.e., S 1 • (See § I for
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Biggs, N. (1974) · doi:10.1017/cbo9780511608704 · Cited by 2,940
Cited by
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Li, M., Liu, R.-R., Lü, L., Hu, M.-B., Xu, S., & Zhang, Y.-C. (2021) · Physics Reports · doi:10.1016/j.physrep.2020.12.003 · Cited by 365
Metadata
- Authors
- Hashimoto, K.-I.
- Venue
- Advanced Studies in Pure Mathematics
- Publisher
- Mathematical Society of Japan
- Published
- 2018-06-20
- Pages
- 211-280
- ISSN
- 0920-1971 (ISSN-L)
- OpenAlex
- W1523808012
- Cited by
- 255
- References
- 1
- FWCI
- 40.02
- Type
- conference paper
- Language
- en
- Open access
- closed
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