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Let R be a non-commutative ring. The commuting graph of R denoted by (R), is a graph with vertex set R \ Z(R), and two distinct vertices a and b are adjacent if ab = ba. In this paper we investigate some properties of (R), whenever R is a finite semisimple ring. For any finite field F, we obtain minimum degree, maximum degree and clique number of (M n (F )). Also it is shown that for any two finite semisimple rings R and S, if (R) (S), then there are commutative semisimple rings R 1 and S 1 and semisimple ring T such that R T × R 1 , S T × S 1 and |R 1 | = |S 1 |.

In this paper we prove a sufficient condition for the existence of matchings in arbitrary groups and its linear analogue, which lead to some generalizations of the existing results in the theory of matchings in groups and central extensions of division rings. We introduce the notion of relative matchings between arrays of elements in groups and use this notion to study the behavior of matchable sets under group homomorphisms. We also present infinite families of prime numbers p such that Z/pZ does not have the acyclic matching property. Finally, we introduce the linear version of acyclic matching property and show that purely transcendental eld extensions satisfy this property.

Abstract. In this paper we study the variation of the p-Selmer rank parities of p-twists of a self-dual Abelian variety over an arbitrary number field K and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to Abelian varieties with full K-rational p-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich-Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

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