Clusters' size-degree distribution for bond percolation

To address some physical properties of percolating systems it can be useful to know the degree distributions in finite clusters along with their size distribution. Here we show that to achieve this aim for classical bond percolation one can use the $q \to 1$ limit of suitably modified q-state Potts model. We consider a version of such model with the additional complex variables and show that its partition function gives generating function for the size and degree distribution in this limit. We derive this distribution analytically for bond percolation on Bethe lattice and complete graph. The possibility to expand the applications of present method to other clusters' characteristics and to models of correlated percolation is discussed.