A proof of Terras' conjecture on the radius of convergence of the Ihara zeta function

We prove the conjecture by Audrey Terras (2011) on the inequalities among the spectral radius ρX of a finite graph X, the radius of convergence R of its Ihara zeta function ZX(u), and the average degree dX¯ of X. Relating to Terras' conjecture, we propose a new conjecture between R and certain Rayleigh quotients.