  Let $G_S$ be the graph obtained by attaching a self-loop at every vertex in
$S \subseteq V(G)$ of a simple graph $G$ of order $n.$ In this paper, we
explore several new results related to the line graph $L(G_S)$ of $G_S.$
Particularly, we show that every eigenvalue of $L(G_S)$ must be at least $-2,$
and relate the characteristic polynomial of the line graph $L(G)$ of $G$ with
the characteristic polynomial of the line graph $L(\widehat{G})$ of a self-loop
graph $\widehat{G}$, which is obtained by attaching a self-loop at each vertex
of $G$. Then, we provide some new bounds for the eigenvalues and energy of
$G_S.$ As one of the consequences, we obtain that the energy of a connected
regular complete multipartite graph is not greater than the energy of the
corresponding self-loop graph. Lastly, we establish a lower bound of the
spectral radius in terms of the first Zagreb index $M_1(G)$ and the minimum
degree $\delta(G),$ as well as proving two Nordhaus-Gaddum-type bounds for the
spectral radius and the energy of $G_S,$ respectively.
