One more remark on the adjoint polynomial

The adjoint polynomial of G is h(G,x)=∑k=1n(−1)n−kak(G)xk,where ak(G) denotes the number of ways one can cover all vertices of the graph G by exactly k disjoint cliques of G. In this paper we show the adjoint polynomial of a graph G is a simple transformation of the independence polynomial of another graph Ĝ. This enables us to use the rich theory of independence polynomials to study the adjoint polynomials. In particular we give new proofs of several theorems of R. Liu and P. Csikvári.