A differential approach for bounding the index of graphs under perturbations

TL;DR

This paper develops bounds for how the spectral radius of a graph changes under various local modifications, using differential inequalities and minimal information about vertex degrees, with characterizations of equality cases.

Contribution

It introduces a differential approach to bound spectral radius variations under graph perturbations, improving understanding of spectral stability with minimal data.

Findings

Derived bounds for spectral radius after vertex/edge additions

Characterized cases where bounds are tight

Discussed asymptotic behavior of bounds

Abstract

This paper presents bounds for the variation of the spectral radius $\lambda(G)$ of a graph $G$ after some perturbations or local vertex/edge modifications of $G$. The perturbations considered here are the connection of a new vertex with, say, $g$ vertices of $G$, the addition of a pendant edge (the previous case with $g=1$) and the addition of an edge. The method proposed here is based on continuous perturbations and the study of their differential inequalities associated. Within rather economical information (namely, the degrees of the vertices involved in the perturbation), the best possible inequalities are obtained. In addition, the cases when equalities are attained are characterized. The asymptotic behavior of the bounds obtained is also discussed. For instance, if $G$ is a connected graph and $G_u$ denotes the graph obtained from $G$ by adding a pendant edge at vertex $u$ with degree $\delta_u$, then, $$ \lambda(G_u)\le \lambda(G)+\frac{\delta_u}{\lambda^3(G)}+\textrm{o}(\frac{1}{\lambda^3(G)}). $$