An Infinite Family of Graphs with the Same Ihara Zeta Function

In 2009, Cooper presented an infinite family of pairs of graphs which were conjectured to have the same Ihara zeta function. We give a proof of this result by using generating functions to establish a one-to-one correspondence between cycles of the same length without backtracking or tails in the graphs Cooper proposed. Our method is flexible enough that we are able to generalize Cooper’s graphs, and we demonstrate additional families of pairs of graphs which share the same zeta function.