On H-joins of complex unit gain graphs and their stability

Let H be a graph of order k and let F = {G1, G2, ..., Gk} be a family of vertex-disjoint graphs. Then the H-join of the family F is obtained by replacing each vertex vi of H with the graph Gi of F and preserving the adjacencies existing in H. This article presents two distinct definitions of the H-join of complex unit gain graphs. The first is a direct extension of the H-join of signed graphs and, as in that setting, is switching-stable only with respect to H. The second, on the other hand, requires the fixing of a spanning forest in order to be defined but has the advantage of being stable under switchings of both H and the component Gi graphs. When applied to signed graphs this provides a new definition of the H-join which does not coincide with the usual version and has the stronger property of being switching-stable with respect to all of the graphs involved. A spectral analysis is provided for both constructions.