M. Fiedler and R. Merris, Irreducibility of associated matrices, Linear Algebra Appl. 37 (1981), 1-10. Math. Reviews 83a:15018.

For an n-by-n matrix A, let K(A) be the associated matrix corresponding to a permutation group of degree m and one of its irreducible characters. Let Dr(A) be the coefficient of the (m-r)th power of x in K(A+xI). If A is irreducible and the character is the principal (identically 1) character, then D1(A) is irreducible. Other results along these lines are proved.

Known as the first (additive) derivation operator on the symmetry class of tensors arising from the given group and character, the mapping that sends A to D1(A) is linear.

As pointed out in R. Merris, Derivations on symmetry classes of tensors and the graph of graphs, Abstracts 4 (1983), 198, if T is the 2-by-2 matrix obtained from the identity by switching its rows then the graph of D1(T) is isomorphic to the graph of graphs. Here, D1 corresponds to the principal character of the pair group, and the graph of graphs has for its vertex set the nonisomorphic graphs on n vertices, where two "vertices" are adjacent if and only if the corresponding graphs on n vertices differ (up to isomorphism) by the addition or deletion of a single edge.