An Optimization Approach to Degree Deviation and Spectral Radius

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 $\hat{I}»$, minimum degree at least $\hat{I}´$, and maximum degree at most $\hat{I}”$, where $0\le \hat{I}´<d<\hat{I}”<n$, we show $$s\leq \frac{2n(Δ-d)(d-δ)}{Δ-δ} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, λ\geq \begin{cases} \frac{d^2n}{\sqrt{d^2n^2-s^2}} & \mbox{, if } s\leq \frac{dn}{\sqrt{2}},\\[3mm] \frac{2s}{n} & \mbox{, if } s> \frac{dn}{\sqrt{2}}. \end{cases}$$ Our results are based on a smoothing technique relating the degree deviation and the largest eigenvalue to low-dimensional non-linear optimization problems.