Dense graphs and sparse matrices

We consider rigorous definitions for dense graphs and sparse matrices, thus quantifying these concepts that have been hitherto used in a qualitative manner. We assign to every graph the compactness index rho, which is a measure of density of graphs and binary matrices. This index takes values rho > 1 for dense graphs and rho < 1 for sparse graphs (matrices). The index is derived from the quotient of the relative graph density and the percentage of zero entries in the adjacency matrix. The numerical values for the compactness index are reported for miscellaneous graphs, including several families of structurally related graphs.