  Let $G_S$ be a graph with loops obtained from a graph $G$ of order $n$ and
loops at $S \subseteq V(G)$. In this paper, we establish a neccesary and
sufficient condition on the bipartititeness of a connected graph $G$ and the
spectrum Spec($G_S$) and Spec($G_{V(G)\backslash S}$). We also prove that for
every $S \subseteq V(G)$, $E(G_S) \geq E(G)$ when $G$ is bipartite. Moreover,
we provide an identification of the spectrum of complete graphs $K_n$ and
complete bipartite graphs $K_{m,n}$ with loops. We characterize any graphs with
loops of order n whose eigenvalues are all positive or non-negative, and also
any graphs with a few distinct eigenvalues. Finally, we provide some bounds
related to $G_S$.
