In this book we mainly present the recent advance in discrete mathematics. Binding number condition for fractional (gfn''m)-critical covered graph and fractional (gfn''m)-critical uniform graph are given characteristics analysis of dendrimers is presented by topological index computing distance-based indices and polynomials for armchair polyhex nanotubes and nanotori are determined. Furthermore we study the reverse eccentric connectivity index of haphthylenic lattices.