On the spectral distribution of distance-\(k\) graph of free product graphs (Q2828066)

For a given graph \(G\), its distance-\(k\) graph \(G^{[k]}\) is defined on the same vertex set by making adjacent those pairs of vertices that are at distance \(k\) in \(G\). The authors are interested in the distribution of spectral properties of distance-\(k\) graphs in the vacuum state. In particular, they describe distributions of the adjacency matrices of distance-\(k\) graphs of \(d\)-regular tree and \(N\)-fold free powers of graphs and the expected eigenvalue distribution of the distance-\(k\) graph of random regular graphs.