  To any nodal curve $C$ is associated the degree class group, a combinatorial
invariant which plays an important role in the compactification of the
generalised Jacobian of $C$ and in the construction of the N\'eron model of the
Picard variety of families of curves having $C$ as special fibre. In this paper
we study this invariant. More precisely, we construct a wide family of graphs
having cyclic degree class group and we provide a recursive formula for the
cardinality of the degree class group of the members of this family. Moreover,
we analyse the behaviour of the degree class group under standard geometrical
operations on the curve, such as the blow up and the normalisation of a node.
