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

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发表日期：

2024年5月28日

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Saieed Akbari,
Irena M. Jovanović,
Johnny Lim

摘要

Let $G_S$ $G_{S}$ be the graph obtained by attaching a self-loop at every vertex in $S \subseteq V(G)$ $S \subseteq V (G)$ of a simple graph G of order n. In this paper, we explore several new results related to the line graph $L(G_S)$ $L (G_{S})$ of $G_S.$ $G_{S} .$ Particularly, we show that every eigenvalue of $L(G_S)$ $L (G_{S})$ must be at least $-2,$ $- 2,$ and relate the characteristic polynomial of the line graph L(G) of G with the characteristic polynomial of the line graph $L({\widehat{G}})$ $L (\hat{G})$ of a self-loop graph ${\widehat{G}}$ $\hat{G}$ , 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 $G_S.$ $G_{S} .$ 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 $M_1(G)$ $M_{1} (G)$ and the minimum degree $\delta (G),$ $δ (G),$ as well as proving two Nordhaus–Gaddum-type bounds for the spectral radius and the energy of $G_S,$ $G_{S},$ respectively.