For a given permutation π n in S n , a random permutation graph is formed by including an edge between two vertices i and j if and only if (i − j)(π n (i) − π n (j)) < 0. In this paper, we study various statistics of random permutation graphs. In particular, we prove central limit theorems for the number m-cliques and cycles of size at least m. Other problems of interest are on the number of isolated vertices, the distribution of a given node (the mid-node as a special case) and extremal degree statistics. Besides, we introduce a directed version of random permutation graphs, and provide two distinct paths that provide variations/generalizations of the model discussed in this paper.

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In this chapter, the authors present a simple model to determine the optimal choice of vaccination scheduling for a society composed of two groups of individuals in order to minimize the economic loss only, assuming herd immunity. First, a simple classical SIR model is presented to form the basis of the analysis; second, the model is revised to include the effects of vaccination which in turn will be extended to include two heterogeneous groups of individuals forming a society. The solutions of relevant differential equations will then be used to calculate the total economic cost of each scenario presented.