Distance between the spectra of certain graphs

Let $$G_n$$ and $$G'_{n}$$ be two nonisomorphic graphs on n vertices with spectra $$\begin{aligned} \lambda _{1} \geqslant \lambda _{2} \geqslant \cdots \geqslant \lambda _{n} \ \ \mathrm {and} \ \ \lambda '_{1} \geqslant \lambda '_{2} \geqslant \cdots \geqslant \lambda '_{n}, \end{aligned}$$ respectively. Define the distance between the spectra of $$G_{n}$$ and $$G'_{n}$$ as $$\begin{aligned} \lambda (G_{n}, G'_{n})= \sum \limits _{i = 1}^n (\lambda _{i}- \lambda _{i}')^{2} \ \mathrm { \big( or \ use \ } \sum \limits _{i = 1}^n \vert \lambda _{i}- \lambda _{i}' \vert \big). \end{aligned}$$ Let $$\begin{aligned} \mathrm {cs}(G_{n})= \mathrm {min} \{ \lambda (G_n, G'_{n}): G'_{n} \ \mathrm { not \ isomorphic \ to} \ G_{n} \}, \end{aligned}$$ and $$\begin{aligned} \text{cs}_{n}=\max \{\text{cs}(G_{n}): G_{n} \text{ a graph on } n \text{ vertices}\}. \end{aligned}$$ Richard Brualdi in [9] proposed the following problems: Problem A. Investigate cs( $$G_{n}$$ ) for special classes of graphs. Problem B. Find a good upper bound on $$\mathrm {cs}_n$$ . In this paper, we investigate problem A and determine cs( $$G_{n}$$ ), when $$G_n$$ is a graph of order n with size two or three. Let $$K_{n}+ K_{1}$$ be a disjoint union of the complete graph $$K_n$$ with one isolated vertex, and $$K_{n+1}^{1}$$ be a complete graph $$K_{n+1}$$ by deleting $$n-1$$ edges from one vertex of $$K_{n+1}$$ . We show that if $$\lambda (K_{n}+ K_{1}, G_{n}')=$$ cs( $$G_{n}$$ ), then $$G_{n}'$$ is isomorphic to $$K_{n+1}^{1}$$ .