Orthogonal polynomials through complex matrix graph theory

We have employed direct complex matrix graph theory techniques to generate orthogonal polynomials such as Hermite polynomials, Chebyshev polynomials of both kinds and Laguerre polynomials of any order without invoking recursion. The techniques involve complex adjacency matrices of graphs and direct computations of their characteristic polynomials, thus providing an elegant technique for the direct computation of these polynomials to high degrees and all roots of these polynomials. Furthermore a tree pruning technique is developed for rapid computations of orthogonal polynomials associated with dendrimers or clustered complete complex-weighted graphs connected by bridges. Tables of several orthogonal polynomials up to degrees 160 are explicitly constructed through these robust complex graph matrix methods. Applications to chemical problems are pointed out.