Operations on Graphs Increasing Some Graph Parameters

In this partly expository paper we discuss and describe some of our old and recent results on partial orders on the set G n of graphs with n vertices and m edges and some operations on graphs within G n that are monotone with respect to these partial orders. The partial orders under consideration include those related with some Laplacian characteristics of graphs as well as with some probabilistic characteristics of graphs with randomly deleted edges. Section 3 contains some basic facts on the Laplacian polynomial of a graph. Section 4 describes various graph operation and their properties. In Section 5 we introduce some partial orders on G n related with the graph Laplacian and the graph reliability (Laplacian posets and reliability posets). Section 6 contains some old and recent results on the -monotonicity of some graph operations with respect to Laplacian posets. Section 7 includes some old and recent results on the -monotonicity of some graph operations with respect to reliability posets and some open problems. In Section 8 we consider some other parameters of graphs and establish some results on -monotonicity of our graph operations with respect to the linear orders on G n related with these parameters. The list of these parameters includes the numbers of Hamiltonian cycles or paths and the numbers of forests of special type. Section 9 contains some generalizations of the described results to weighted graphs.