The discrete-time quaternionic quantum walk and the second weighted zeta function on a graph

We define the quaternionic quantum walk on a finite graph and investigate its properties. This walk can be considered as a natural quaternionic extension of the Grover walk on a graph. We explain the way to obtain all the right eigenvalues of a quaternionic matrix and a notable property derived from the unitarity condition for the quaternionic quantum walk. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using complex eigenvalues of the quaternionic weighted matrix which is easily derivable from the walk. Since our derivation is owing to a quaternionic generalization of the determinant expression of the second weighted zeta function, we explain the second weighted zeta function and the relationship between the walk and the second weighted zeta function.Comment: 15 page

On Auslander's n-gorenstein rings

AbstractAccording to Auslander, a Noetherian ring R is called n-Gorenstein for n ≥ 1 if in a minimal injective resolution 0 → RR → E0 → E1 → … → En →, …, the flat dimension of each Ei is at most i for i = 0, 1, …, n − 1. We prove that for an n-Gorenstein ring R of self-injective dimension n, the last term En in a minimal injective resolution of RR has essential socle.We also prove that the 1-Gorenstein property is inherited by a maximal quotient ring, and as a related result, we characterize a Noetherian ring of dominant dimension at least 2