An Optimization Approach to Degree Deviation and Spectral Radius

TL;DR

This paper develops an optimization-based method to relate degree deviation and spectral radius in graphs, providing bounds and insights into their interplay for graphs with given degree constraints.

Contribution

It introduces a novel smoothing technique that connects degree deviation and spectral radius through low-dimensional non-linear optimization, offering new bounds and analysis.

Findings

Derived an upper bound for degree deviation in graphs.

Established lower bounds for the largest eigenvalue based on degree deviation.

Introduced a smoothing technique linking spectral properties to degree distribution.

Abstract

For a finite, simple, and undirected graph $G$ with $n$ vertices and average degree $d$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-d\right|$. Provided that $G$ has largest eigenvalue $\lambda$, minimum degree at least $\delta$, and maximum degree at most $\Delta$, where $0\leq\delta<d<\Delta<n$, we show $$s\leq \frac{2n(\Delta-d)(d-\delta)}{\Delta-\delta} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \lambda \geq \begin{cases} \frac{d^2n}{\sqrt{d^2n^2-s^2}} & \mbox{, if } s\leq \frac{dn}{\sqrt{2}},\\[3mm] \frac{2s}{n} & \mbox{, if } s> \frac{dn}{\sqrt{2}}. \end{cases}$$ Our results are based on a smoothing technique relating the degree deviation and the largest eigenvalue to low-dimensional non-linear optimization problems.