In this paper, only simple graphs are considered. A graphG is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G satisfies reciprocal eigenvalue property (property R) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property (property SR) if the reciprocal of each eigenvalue o...

Here is an explicit question: are all curves (say reduced, even non-singular, finite type over given k) isomorphic? Obviously not: some are affine, and some (such as P) are not. So to simplify things — and we’ll further motivate this simplification in Class 42 — are all projective curves isomorphic? Perhaps all nonsingular projective curves are isomorphic to P? Once again the answer is no, but ...

in this paper, we study some ring theoretic properties of the amalgamated duplication ring $rbowtie i$ of a commutative noetherian ring $r$ along an ideal $i$ of $r$ which was introduced by d'anna and fontana. indeed, it is determined that when $rbowtie i$ satisfies serre's conditions $(r_n)$ and $(s_n)$, and when is a normal ring, a generalized cohen-macaulay ring and finally a filter ring.