Parameters of connectivity in (a,b)-linear graphs

For some a and b positive rational numbers, a simple graph with n vertices and m=an−b edges is an (a,b)-linear graph, when n>2b. We characterize non-empty classes of (a,b)-linear graphs and determine those which contain connected graphs. For non-empty classes, we build sequences of (a,b)-linear graphs and sequences of connected (a,b)-linear graphs. Furthermore, for each of these sequences where every graph is bounded by a constant, we show that its correspondent sequence of diameters diverges, while its correspondent sequence of algebraic connectivities converges to zero.