Graph Polynomials and Graph Transformations in Algebraic Graph Theory

The thesis consists of two parts. In the first part we study two graph transformations,namely the Kelmans transformation and the generalized tree shift. In the second part of thisthesis we study an extremal graph theoretic problem and its relationship with algebraic graphtheory. The main results of this thesis are the following. • We show that the Kelmans transformation is a very effective tool in many extremal alge-braic graph theoretic problems. Among many other things, we attain a breakthrough ina problem of Eva Nosal by the aid of this transformation.• We define the generalized tree shift which turns out to be a powerful tool in many extremalgraph theoretic problems concerning trees. With the aid of this transformation we provea conjecture of V. Nikiforov. We give a strong method for attacking extremal graphtheoretic problems involving graph polynomials and trees. By this method we give newproofs for several known results and we attain some new results.• We completely solve the so-called density Turán problem for trees and we give sharpbounds for the critical edge density in terms of the largest degree for every graphs. Weestablish connection between the problem and algebraic graph theory. By the aid of thisconnection we construct integral trees of arbitrarily large diameters. This was an openproblem for more than 30 years.