Combinatorial structures associated with finite-dimensional Leibniz algebras M. Ceballos † , J. Núñez † , Á. F. Tenorio ‡ 1 † Departamento de Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla. Aptdo. 1160. 41080-Seville (Spain). ‡ Dpto. de Economía, Métodos Cuantitativos e Historia Económica. Escuela Politécnica Superior. Universidad Pablo de Olavide. Ctra. Utrera km. 1. 41013-Seville (Spain). mceballos@us.es jnvaldes@us.es aftenorio@upo.es Abstract This paper is the second in a series giving the foundations for associating Leibniz algebras with combinatorial structures of dimension 2. On this occasion, we character- ize Leibniz algebras of dimension greater than 3 associated with (psuedo)digraphs and, consequently, we also study the converse, proving some properties about (pseudo)digraphs associated with Leibniz algebras. Finally, we analyze the nilpotency and solvability for Leibniz algebras associated with (pseudo)digraphs. Keywords: (Pseudo)digraph, Combinatorial structure, Leibniz algebra. 2010 Mathematics Subject Classification: 17A32, 17A60, 05C25, 05C20, 05C90. 1 Introduction In Mathematics, discovering and researching links and relations between different fields, is very stimulating. Thanks to these connections, researchers can solve open problems by means of alternative tools and procedures, leading to new properties and results as well as introducing new theories. In this paper, we deal with the relation between Graph Theory and Leibniz algebras expounded in [6]. More concretely, the main purpose consists in carrying on the research started in [6] in order to set a translator for properties on Leibniz algebras to combinatorial structures and vice versa in the sense indicated in [2] for Lie algebras. The quest for this generalization is based on the fact that Leibniz algebras (introduced by Loday [9] at the beginning of the 1990s) are non-associative algebras which provide us with a non-commutative generalization of Lie algebras, being the latter a particular case of the first. In this way, an important issue of Leibniz algebras consists in the possibility of extending many well-known results on Lie algebras. For example, the right-multiplication operator on an element of a Leibniz algebra is a derivation, as happened for Lie algebras. 1 Corresponding author. Phone number: +34-954349345. Fax number: +34-954349339 1