Line graphs and Nordhaus-Gaddum-type bounds for self-loop graphs

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 $n\mathrm{.}$ In this paper, we explore several new results related to the line graph $L(G_S)$ of $G_S\mathrm{.}$ 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 $G_S\mathrm{.}$ 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.