One more remark on the adjoint polynomial

The adjoint polynomial of $G$ is $$h(G,x)=\sum_{k=1}^n(-1)^{n-k}a_k(G)x^k,$$ where $a_k(G)$ 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 the adjoint polynomial of a graph $G$ is a simple transformation of the independence polynomial of another graph $\widehat{G}$. This enables us to use the rich theory of independence polynomials to study the adjoint polynomials. In particular we a give new proofs of several theorems of R. Liu and P. Csikv\'{a}ri.