Line Graphs and Nordhaus–Gaddum-Type Bounds for Self-Loop Graphs

Abstract

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.