  This paper deals with conservation laws on networks, represented by graphs.
  Entropy-type conditions are considered to determine dynamics at nodes.
  Since entropy dispersion is a local concept, we consider a network composed
by a single node $J$ with $n$ incoming and $m$ outgoing arcs. We extend at $J$
the classical Kru\v{z}kov entropy obtaining two conditions, denoted by (E1) and
(E2): the first requiring entropy condition for all Kru\v{z}kov entropies, the
second only for the value corresponding to sonic point. First we show that in
case $n \ne m$, no Riemann solver can satisfy the strongest condition. Then we
characterize all the Riemann solvers at $J$ satisfying the strongest condition
(E1), in the case of nodes with at most two incoming and two outgoing arcs.
Finally we focus three different Riemann solvers, introduced in previous
papers. In particular, we show that the Riemann solver introduced for data
networks is the only one always satisfying (E2).
