In mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix. The Ihara zeta function was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups of the two-by-two p-adic special linear group. Jean-Pierre Serre suggested in his book Trees that Ihara's original definition can be reinterpreted graph-theoretically. It was Toshikazu Sunada who put this suggestion into practice in 1985. As observed by Sunada, a regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis.[1]
The Ihara zeta function is defined as the analytic continuation of the infinite product
\zetaG(u)=\prodp
| 1 | |
| 1-u{L(p) |
where L(p) is the length
L(p)
p
p
G=(V,E)
p
G
p=(v0,\ldots,vk-1)
(vi,v(i+1)\bmod)\inE,
vi ≠ v(i+2).
k
L(p)
p
m
m>1
This graph-theoretic formulation is due to Sunada.
Ihara (and Sunada in the graph-theoretic setting) showed that for regular graphs the zeta function is a rational function.If
G
q+1
A
\zetaG(u)=
| 1 | |
| (1-u2)r(G)-1\det(I-Au+qu2I) |
,
where
r(G)
G
G
n
r(G)-1=(q-1)n/2
The Ihara zeta-function is in fact always the reciprocal of a graph polynomial:
\zetaG(u)=
| 1 | |
| \det(I-Tu) |
~,
where
T
The Ihara zeta function plays an important role in the study of free groups, spectral graph theory, and dynamical systems, especially symbolic dynamics, where the Ihara zeta function is an example of a Ruelle zeta function.[3]
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. Toshikazu Sunada . Curvature and Topology of Riemannian Manifolds . 1201 . 1986 . 266–284 . 10.1007/BFb0075662 . L-functions in geometry and some applications . . 978-3-540-16770-9 . 0605.58046 .
. Harold Stark . Multipath zeta functions of graphs . 601–615 . Emerging Applications of Number Theory . Dennis A. . Hejhal . Dennis Hejhal . Joel . Friedman . Martin C. . Gutzwiller . Martin Gutzwiller . Andrew M. . 3 . Odlyzko. Andrew Odlyzko . . 1999 . 0-387-98824-6 . 0988.11040 . IMA Vol. Math. Appl. . 109 .
. Audrey Terras . A survey of discrete trace formulas . 643–681 . Emerging Applications of Number Theory . Dennis A. . Hejhal . Dennis Hejhal . Joel . Friedman . Martin C. . Gutzwiller . Martin Gutzwiller . Andrew M. . 3 . Odlyzko. Andrew Odlyzko . Springer . 1999 . 0-387-98824-6 . 0982.11031. IMA Vol. Math. Appl. . 109 .
. Zeta Functions of Graphs: A Stroll through the Garden . 128 . Cambridge Studies in Advanced Mathematics . Audrey Terras . . 2010 . 0-521-11367-9 . 1206.05003 .