  In this paper we introduce new techniques in order to deepen into the
structure of a Leavitt path algebra with the aim of giving a description of the
center. Extreme cycles appear for the first time; they concentrate the purely
infinite part of a Leavitt path algebra and, jointly with the line points and
vertices in cycles without exits, are the key ingredients in order to determine
the center of a Leavitt path algebra. Our work will rely on our previous
approach to the center of a prime Leavitt path algebra \cite{CMMSS1}. We will
go further into the structure itself of the Leavitt path algebra. For example,
the ideal $I(P_{ec} \cup P_{c} \cup P_l)$ generated by vertices in extreme
cycles ($P_{ec}$), by vertices in cycles without exits ($P_c$) and by line
points ($P_l$) will be a dense ideal in some cases, for instance in the finite
one or, more generally, if every vertex connects to $P_l \cup P_c\cup P_{ec}$.
Hence its structure will contain much of the information about the Leavitt path
algebra. In the row-finite case, we will need to add a new hereditary set: the
set of vertices whose tree has infinite bifurcations ($P_{b^\infty}$).
