An Ihara formula for partially directed graphs

Ihara’s formula expresses the Ihara zeta function of a finite undirected graph as a rational function with a particularly nice form. In 2001 Mizuno and Sato showed that the Ihara zeta function of a fully directed graph has a similar expression, and in 2005, Sato generalized Ihara’s formula to connected, simple, partially directed graphs. (Sato proved his formula for the more-general two-variable Bartholdi zeta function.) This paper provides a new proof of Ihara’s formula for the Ihara zeta function of any finite graph, not necessarily connected or simple, no matter whether it is undirected, fully directed, or partially directed.