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]
Definition
The Ihara zeta function is defined as the analytic continuation of the infinite product
ζ
G
(
u
)
=
∏
p
1
1
−
u
L
(
p
)
,
{\displaystyle \zeta _{G}(u)=\prod _{p}{\frac {1}{1-u^{{L}(p)}}},}
where L (p ) is the length
L
(
p
)
{\displaystyle L(p)}
of
p
{\displaystyle p}
.
The product in the definition is taken over all prime
closed geodesics
p
{\displaystyle p}
of the graph
G
=
(
V
,
E
)
{\displaystyle G=(V,E)}
, where geodesics which differ by a
cyclic rotation are considered equal. A closed geodesic
p
{\displaystyle p}
on
G
{\displaystyle G}
(known in graph theory as a "
reduced closed walk "; it is not a graph geodesic) is a finite sequence of vertices
p
=
(
v
0
,
…
,
v
k
−
1
)
{\displaystyle p=(v_{0},\ldots ,v_{k-1})}
such that
(
v
i
,
v
(
i
+
1
)
mod
k
)
∈
E
,
{\displaystyle (v_{i},v_{(i+1){\bmod {k}}})\in E,}
v
i
≠
v
(
i
+
2
)
mod
k
.
{\displaystyle v_{i}\neq v_{(i+2){\bmod {k}}}.}
The integer
k
{\displaystyle k}
is the length
L
(
p
)
{\displaystyle L(p)}
. The closed geodesic
p
{\displaystyle p}
is prime if it cannot be obtained by repeating a closed geodesic
m
{\displaystyle m}
times, for an integer
m
>
1
{\displaystyle m>1}
.
This graph-theoretic formulation is due to Sunada.
Ihara's formula
Ihara (and Sunada in the graph-theoretic setting) showed that for regular graphs the zeta function is a rational function.
If
G
{\displaystyle G}
is a
q
+
1
{\displaystyle q+1}
-regular graph with
adjacency matrix
A
{\displaystyle A}
then
[2]
ζ
G
(
u
)
=
1
(
1
−
u
2
)
r
(
G
)
−
1
det
(
I
−
A
u
+
q
u
2
I
)
,
{\displaystyle \zeta _{G}(u)={\frac {1}{(1-u^{2})^{r(G)-1}\det(I-Au+qu^{2}I)}},}
where
r
(
G
)
{\displaystyle r(G)}
is the
circuit rank of
G
{\displaystyle G}
. If
G
{\displaystyle G}
is connected and has
n
{\displaystyle n}
vertices,
r
(
G
)
−
1
=
(
q
−
1
)
n
/
2
{\displaystyle r(G)-1=(q-1)n/2}
.
The Ihara zeta-function is in fact always the reciprocal of a
graph polynomial :
ζ
G
(
u
)
=
1
det
(
I
−
T
u
)
,
{\displaystyle \zeta _{G}(u)={\frac {1}{\det(I-Tu)}}~,}
where
T
{\displaystyle T}
is Ki-ichiro Hashimoto's edge adjacency operator.
Hyman Bass gave a determinant formula involving the adjacency operator.
Applications
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]
References
^ Terras (1999) p. 678
^ Terras (1999) p. 677
^ Terras (2010) p. 29
Ihara, Yasutaka (1966).
"On discrete subgroups of the two by two projective linear group over
p
{\displaystyle {\mathfrak {p}}}
-adic fields" . Journal of the Mathematical Society of Japan . 18 : 219–235.
doi :
10.2969/jmsj/01830219 .
MR
0223463 .
Zbl
0158.27702 .
Sunada, Toshikazu (1986). "L-functions in geometry and some applications". Curvature and Topology of Riemannian Manifolds .
Lecture Notes in Mathematics . Vol. 1201. pp. 266–284.
doi :
10.1007/BFb0075662 .
ISBN
978-3-540-16770-9 .
Zbl
0605.58046 .
Bass, Hyman (1992). "The Ihara-Selberg zeta function of a tree lattice".
International Journal of Mathematics . 3 (6): 717–797.
doi :
10.1142/S0129167X92000357 .
MR
1194071 .
Zbl
0767.11025 .
Stark, Harold M. (1999). "Multipath zeta functions of graphs". In
Hejhal, Dennis A. ; Friedman, Joel;
Gutzwiller, Martin C. ; et al. (eds.). Emerging Applications of Number Theory . IMA Vol. Math. Appl. Vol. 109.
Springer . pp. 601–615.
ISBN
0-387-98824-6 .
Zbl
0988.11040 .
Terras, Audrey (1999). "A survey of discrete trace formulas". In
Hejhal, Dennis A. ; Friedman, Joel;
Gutzwiller, Martin C. ; et al. (eds.). Emerging Applications of Number Theory . IMA Vol. Math. Appl. Vol. 109. Springer. pp. 643–681.
ISBN
0-387-98824-6 .
Zbl
0982.11031 .
Terras, Audrey (2010). Zeta Functions of Graphs: A Stroll through the Garden . Cambridge Studies in Advanced Mathematics. Vol. 128.
Cambridge University Press .
ISBN
0-521-11367-9 .
Zbl
1206.05003 .