In this paper, we study the independence polynomial $P_G(x)$ of a finite simple graph $G$, with emphasis on the evaluation at $x=-1$, symmetry, and its connection with the $h$-polynomial of the edge ideal of $G$. For big star graphs, we determine exactly when $P_G(-1)$ is $0, 1$, or $-1$, characterize the pseudo-Gorenstein$^*$ members, and show that there is a unique big star with symmetric independence polynomial. We also study graphs obtained from a graph $H$ by attaching leaves to selected vertices. We derive an explicit formula for the resulting independence polynomial, determine the corresponding value at $-1$, and prove that if every vertex of $H$ receives at least one leaf, then the independence polynomial is symmetric if and only if each vertex receives exactly two leaves. As an application, we obtain exact criteria for the values of $P_G(-1)$ and for the pseudo-Gorenstein$^*$ members of caterpillar graphs. For cochordal graphs, we classify all symmetric independence polynomials. Finally, for connected graphs on $n$ vertices with small independence numbers, we determine the exact range of possible values of $P_G(-1)$.