  In this article, we define an information-theoretic entropy based on the
Ihara zeta function of a graph which is called the Ihara entropy. A dynamical
system consists of a billiard ball and a set of reflectors correspond to a
combinatorial graph. The reflectors are represented by the vertices of the
graph. Movement of the billiard ball between two reflectors is represented by
the edges. The prime cycles of this graph generate the bi-infinite sequences of
the corresponding symbolic dynamical system. The number of different prime
cycles of a given length can be expressed in terms of the adjacency matrix of
the oriented line graph. It also constructs the formal power series expansion
of Ihara zeta function. Therefore, the Ihara entropy has a deep connection with
the dynamical system of billiards. As an information-theoretic entropy, it
fulfils the generalized Shannon-Khinchin axioms. It is a weakly decomposable
entropy whose composition law is given by the Lazard formal group law.
