Extremal graphs for the Tutte polynomial

A graph transformation called the compression of a graph G is known to decrease the number of spanning trees, the all terminal reliability, and the magnitude of the coefficients of the chromatic polynomial of a graph G. All of these graph parameters can be derived from the Tutte polynomial of G, and in this paper we determine more generally compression's effect on the Tutte polynomial, recovering the previous results and obtaining similar results for a wide variety of other graph parameters derived from the Tutte polynomial. Since any simple connected graph can be transformed into a connected threshold graph via a series of compressions, this gives that threshold graphs are extremal simple graphs for all of the parameters considered. (c) 2021 Elsevier Inc. All rights reserved.