  In this paper, we introduce the concept of $k$-integral graphs. A graph
$\Gamma$ is called $k$-integral if the extension degree of the splitting field
of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is
equal to $k$. We prove that for any positive integers $k$ and $\Delta$, the set
of all finite connected graphs with algebraic degree at most $k$ and maximum
degree at most $\Delta$ is finite. We study
  $2$-integral Cayley graphs over finite groups $G$ with respect to Cayley sets
which are a union of conjugacy classes of $G$. Among other general results, we
completely characterize all finite abelian groups having a connected
$2$-integral Cayley graph with valency $2,3,4$ and $5$.
  Furthermore, we classify the finite groups $G$ that all Cayley graphs over
$G$ with bounded valency are $2$-integral.
