(\ell^1\)-Cospectrality of graphs

The following problem has been proposed in [Research problems from the Aveiro workshop on graph spectra, {\em Linear Algebra and its Applications}, {\bf 423} (2007) 172-181.]:\\ (Problem AWGS.4) Let \(G_n\) and \(G'_n\) be two nonisomorphic graphs on \(n\) vertices with spectra $$\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n \;\;\;\text{and}\;\;\; \lambda'_1 \geq \lambda'_2 \geq \cdots \geq \lambda'_n,$$ respectively. Define the distance between the spectra of \(G_n\) and \(G'_n\) as $$\lambda(G_n,G'_n) =\sum_{i=1}^n (\lambda_i-\lambda'_i)^2 \;\;\; \big(\text{or use}\; \sum_{i=1}^n|\lambda_i-\lambda'_i|\big).$$ %Let \(\epsilon\) be a nonnegative number. Graphs \(G_n\) and \(G'_n\) are \(\epsilon\)-cospectral if \(\lambda(G_n,G'_n)\leq \epsilon\). Thus, \(G_n\) %and \(G'_n\) are \(0\)-cospectral if and only if \(G_n\) and \(G'_n\) are cospectral. Define the cospectrality of \(G_n\) by $$\text{cs}(G_n) = \min\{\lambda(G_n,G'_n) \;:\; G'_n \;\;\text{not isomorphic to} \; G_n\}.$$ %Thus \(\text{cs}(G_n) = 0\) if and only if \(G_n\) has a cospectral mate. %This function measures how far apart the spectrum of a graph with \(n\) vertices can be from the %spectrum of any other graph with \(n\) vertices.\\ {\bf Problem A.} Investigate \(\text{cs}(G_n)\) for special classes of graphs. In this paper we study Problem A for certain graphs with respect to the \(\ell^1\)-norm, i.e. \(\sigma(G_n,G'_n)=\sum_{i=1}^n|\lambda_i-\lambda'_i|\). We find \(\text{cs}(K_n)\), \(\text{cs}(nK_1)\), \(\text{cs}(K_2+(n-2)K_1)\) (\(n\geq 2\)), \(\text{cs}(K_{n,n})\) and \(\text{cs}(K_{n,n+1})\), where \(K_n, nK_1, K_2+(n-2)K_1, K_{n,m} \) denote the complete graph on \(n\) vertices, the null graph on \(n\) vertices, the disjoint union of the \(K_2\) with \(n-2\) isolated vertices (\(n\geq 2\)), and the complete bipartite graph with parts of sizes \(n\) and \(m\), respectively.