We consider a family of dynamical systems (A,alpha,L) in which alpha is an
endomorphism of a C*-algebra A and L is a transfer operator for \alpha. We
extend Exel's construction of a crossed product to cover non-unital algebras A,
and show that the C*-algebra of a locally finite graph can be realised as one
of these crossed products. When A is commutative, we find criteria for the
simplicity of the crossed product, and analyse the ideal structure of the
crossed product.