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

Let GS be the graph obtained by attaching a self-loop at every vertex in S ⊆ V(G) of a simple graph G of order $n.$ In this paper, we explore several new results related to the line graph L(GS) of GS. Particularly, we show that every eigenvalue of L(GS) 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(G) of a self-loop graph 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 GS. 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 M₁(G) and the minimum degree δ(G), as well as proving two Nordhaus-Gaddum-type bounds for the spectral radius and the energy of GS, respectively.