Let $G$ be a finite simple graph on $n$ vertices and set $R=\Bbbk[x_1,\dots,x_n]$, with edge ideal $I(G)$ and cover ideal $J(G)$. We give an explicit description of the $h$-polynomial of $R/J(G)$, in a form that extends to the Alexander dual of any squarefree monomial ideal. We then express $\textrm{deg } h_{R/I(G)}(t)$ and $\textrm{deg } h_{R/J(G)}(t)$ in terms of the independence polynomial $P_G(x)=\sum_{i\ge 0} g_i x^i$ via an invariant $M(G)$, the multiplicity of $x=-1$ as a root of $P_G(x)$. In particular, we prove \[\textrm{deg } h_{R/I(G)}(t)=α(G)-M(G) \qquad\text{and}\qquad \textrm{deg } h_{R/J(G)}(t)=n-2-M(G), \] where $α(G)$ is the independence number of $G$. As a corollary, $M(G)$ is the additive inverse of the $\mathfrak{a}$-invariants of $R/I(G)$ and $R/J(G)$. We develop recursions and closed formulas for $M(G)$ for broad graph families, and use them to analyze which (reg, deg h)-pairs occur for cover ideals within chordal classes, including explicit constructions realizing extremal behavior. We conclude with a conjectural bound on $\left|\textrm{reg }(R/J(G))-\textrm{deg } h_{R/J(G)}(t)\right|$ for connected graphs.