Distance between the spectra of certain graphs

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