Meghodipa Das

• A classical topic in the theory of random graphs is the probability of at least one isolated vertex in a given random graph. An isolated node has a huge impact on social networks which can be given by a random graph. We present a distribution on the number of isolated vertex using the probability generating function. We discuss the relationship between isolated edges and extended cut polynomials, extended matching polynomials using the principle of inclusion exclusion. We introduce an algorithm based on colored graphs for general graphs. We apply this to the components of a graph as well. Finally, we implement the idea on a special class of graphs like cycle, bipartite graph, path, and others. We discuss recursive procedure based on the analogous coloring rules for ladder and fan graphs.