Co-adjoint polynomial

In this note we study a certain graph polynomial arising from a special recursion. This recursion is a member of a family of four recursions where the other three recursions belong to the chromatic polynomial, the modified matching polynomial, and the adjoint polynomial, respectively. The four polynomials share many common properties, for instance all of them are of exponential type, i. e., they satisfy the identity $$\sum_{S\subseteq V(G)}f(G[S],x)f(G[V\setminus S],y)=f(G,x+y)$$ for every graph $G$. It turns out that the new graph polynomial is a specialization of the Tutte polynomial.