Applications of the Kelmans Transformation: Extremality of the Threshold Graphs

In this paper we study various extremal problems related to some combinatorially defined graph polynomials such as matching polynomial, chromatic polynomial, Laplacian polynomial. It will turn out that many problems attain its extremal value in the class of threshold graphs. To attack these kinds of problems we survey several applications of the so-called Kelmans transformation. Mea culpa. This paper was rewritten after we learnt that we missed several important papers on the Kelmans transformation. We hope that we managed to fix the historical notes. We also grasped the opportunity to make the manuscript more readable.