A combinatorial discussion on ?nit edimensional Leavitt path algebras

Abstract

AnyfinitedimensionalsemisimplealgebraAoverafieldKisisomorphictoadirectsumoffinitedimensionalfullmatrixringsoversuitabledivisionrings.Weshallconsiderthedirectsumoffinitedimensionalfullmatrixringsoverafield K. AllsuchfinitedimensionalsemisimplealgebrasariseasfinitedimensionalLeavittpathalgebras.Forthisspecificfinitedimensionalsemisimplealgebra A overafield K, wedefineauniquelydeterminedspecificgraph-calledatruncatedtreeassociatedwith A -whoseLeavittpathalgebraisisomorphicto A .Wedefineanalgebraicinvariant κ( A ) for A andcountthenumberofisomorphismclassesofLeavittpathalgebraswiththesamefixedvalueof κ(A ).Moreover,wefindthemaximumandtheminimum K-dimensionsoftheLeavittpathalgebrasofpossibletreeswithagivennumberofverticesandwealsodeterminethenumberofdistinctLeavittpathalgebrasoflinegraphswithagivennumberofvertices.