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

TL;DR

This paper investigates the spectral properties of self-loop graphs, deriving bounds on eigenvalues, energy, and spectral radius, and relating these to classical graph invariants and Nordhaus-Gaddum-type bounds.

Contribution

It introduces new spectral bounds for self-loop graphs, relates characteristic polynomials of line graphs with and without self-loops, and extends Nordhaus-Gaddum bounds to these graphs.

Findings

Eigenvalues of line graphs of self-loop graphs are at least -2.

Energy of regular complete multipartite graphs is bounded by that of their self-loop counterparts.

New lower bounds for spectral radius involving Zagreb index and minimum degree.

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.