The graph zeta function began from the Ihara zeta function. We state the history and the definition of the Ihara zeta function and explain its properties. Next, we introduce the second weighted zeta function of a graph as a generalization of the Ihara zeta function and provide its determinant expression. As an application of the second weighted zeta function, we explain an explicit formula, i.e., the Konno–Sato Theorem for the time evolution matrix (the Grover matrix) of the Grover walk as a discrete-time quantum walk on a graph. Finally, as an application of the Konno–Sato Theorem, we briefly discuss Grover/Zeta Correspondence, which makes an initial move of a series of Zeta Correspondence.
