# $G$-analogues of symmetric functions (reference request)

Let $$G$$ be a simple graph with vertex set $$V$$. Stanley defined the $$G$$-analogs of the symmetric function as follows: For $$i \ge 0$$, define $$e^G_i = \sum_S \big(\prod_{v \in S}v\big)$$ where the sum runs over independent /stable subsets of size $$i$$ in $$G$$.

Since the classical elementary symmetric functions $$e_{i}$$ form a basis for the ring of symmetric functions $$e_{i}$$ goes to $$e^G_{i}$$ defines a homomorphism from ring of symmetric functions to $$\mathbb{Z}(V)$$. Using this homomorphism, we can define $$G$$-analogs of other classical symmetric functions also.