Parameters of connectivity in ()-linear graphs

For some a and b positive rational numbers, a simple graph with n vertices and m = a n − b edges is an ( a , b ) -linear graph, when n > 2 b . 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.