On Reciprocal Distance Spectrum of Neighbourhood Corona of Graphs

The reciprocal distance spectrum of a connected graph G is the set of all eigenvalues of its reciprocal distance matrix. The neighbourhood corona \(G \star H\) of two graphs G and H is formed by taking one copy of G and |V(G)| copies of H and making all the vertices in the \(i{\text {th}}\) copy of H adjacent with all the neighbours of the \(i{\text {th}}\) vertex in G. In this paper we describe the reciprocal distance eigenvalues and corresponding eigenvectors of \(G \star H\) in terms of the adjacency eigenvalues of G and H when G is a regular triangle-free graph of diameter 2 and H is regular. As applications we construct infinitely many reciprocal distance non-cospectral pairs of reciprocal distance equienergetic graphs and non-isomorphic pairs of reciprocal distance cospectral graphs of the same order and size using line graphs, iterated line graphs, double graphs, strong double graphs, and complement graphs.