Random Networks, Graphs, and Matrices

The theory of elasticity based on the distribution function of the gyration tensor is reviewed. It is shown that the James-Guth potential for a network is a model potential of mean force, derivable in principle from the true many-body potential for the elastic body. The determination of the modulus of elasticity is resolved in the calculation of the statistical mechanical average of the smallest, non-zero, eigenvalue of the connectivity matrix that describes the network. The mathematical problem to determine the spectrum of eigenvalues of the random network can be couched in both the language of random graphs and of random matrices.