Summary: {\it S. Negami} and {\it K. Kawagoe} [Discrete Appl. Math. 56, No. 2-3, 323-331 (1995; Zbl 0842.05091)] have already defined a polynomial $\widetilde{f} (G)$ associated with each graph $G$ as what discriminates graphs more finely than the polynomial $f(G)$ defined by {\it S. Negami} [Polynomial invariants of graphs, Trans. Am. Math. Soc. 299, 601-622 (1987; Zbl 0674.05062)] and the Tutte polynomial. In this paper, we show that the polynomial $\widetilde{f} (G)$ includes potentially the generating function counting the independent sets and the degree sequence of a graph $G$, which cannot be recognized from $f(G)$ in general, and discuss on $\widetilde{f} (T)$ of trees $T$ with observations by computer experiments.