The Number of Mappings of Graphs, Ordering of Graphs, and Muirhead's Theorem

The following ordering of graphs is introduced: we say that a graph $D_1$ is greater than a graph $D_2$ if for any graph $\Gamma$ the number of mappings of the graph $D_1$ to the graph $\Gamma$ is not less than the number of mappings of the graph $D_2$ to the graph $\Gamma$. We prove a number of theorems that allow comparison of some graphs. An interesting relationship of this problem with the theory of homogeneous polynomials is established, in particular with Muirhead's well-known theorem.