Bass' identity and a coin arrangements lemma

There is no doubt that the Riemann zeta function is a function of great significance in number theory because of its connection to the distribution of prime numbers. The Ihara zeta function was first introduced by Ihara [5]. This zeta function is associated with a finite graph and aperiodic closed walks in it. Bass’ identity [1] shows that the Ihara zeta function is a determinant. It is shown in the book [12] that using this identity and some other ideas from linear algebra one can come up with the graph-theoretical analogue of the prime number theorem. Therefore, it seems that Bass’ identity plays an essential role in the theory of the Ihara zeta function of a finite graph. Since the original work of Bass [1], his identity has been proven several times in an inspiringway. Let usmention for instance [3] and a lecture of Xavier Viennot at the Newton Institute, Cambridge University in 2008. Independently, an evaluation of the Ihara zeta function of planar graphs and a determinantal formula for it appeared implicitly in thework of Kac,Ward, Feynman and Sherman on the 2-dimensional Ising problem [6,10]. This work is thus an important special case of Bass’ identity. This combinatorial approach to the Ising problem has been less popular among combinatorialists and physicists than the Pfaffian method (see [7] for an introduction to both the Feynman and the Pfaffian methods as well as the relation with the Ihara zeta function and Bass’ identity). Quite recently, however Feynman’s approach reappeared in studies of the criticality of the Ising problem on closed Riemann surfaces of a positive genus [8]. New studies of the Feynman method as well as comparisons with the Pfaffian method appear (see e.g. [2]).