Spectral behavior of some graph and digraph compositions

Let G be a graph of order n the vertices of which are labeled from 1 to n and let $G_1$, · · · ,$G_n$ be n graphs. The graph composition G[$G_1$, · · · ,$G_n$] is the graph obtained by replacing the vertex i of G by the graph Gi and there is an edge between u ∈ $G_i$ and v ∈ $G_j$ if and only if there is an edge between i and j in G. We first consider graph composition G[$K_k$, · · · ,$K_k$] where G is regular and $K_k$ is a complete graph and we establish some links between the spectral characterisation of G and the spectral characterisation of G[$K_k$, · · · ,$K_k$]. We then prove that two non isomorphic graphs G[$G_1$, · · ·$G_n$] where $G_i$ are complete graphs and G is a strict threshold graph or a star are not Laplacian-cospectral, giving rise to a spectral characterization of these graphs. We also consider directed graphs, especially the vertex-critical tournaments without non-trivial acyclic interval which are tournaments of the shape t[$\overrightarrow{C}_{k_1}$, · · · ,$\overrightarrow{C}_{k_m}$], where t is a tournament and $\overrightarrow{C}_{k_i}$ is a circulant tournament. We give conditions to characterise these graphs by their spectrum.