On the reciprocal distance spectrum of edge corona of graphs

The reciprocal distance spectrum (Harary spectrum) of a connected graph [Formula: see text] is the multiset of eigenvalues of its reciprocal distance matrix (Harary matrix) [Formula: see text], which plays a crucial role in spectral graph theory and finds applications in chemistry in studying the physical properties of chemical compounds by considering the associated molecular graph. The reciprocal distance matrix and its eigenvalues are closely connected to various graph invariants and parameters. The reciprocal distance energy (Harary energy) of [Formula: see text] is the sum of the absolute values of the reciprocal distance eigenvalues of [Formula: see text], which is an essential concept in chemical graph theory to study the physico-chemical properties of compounds. Graph operations are essential in graph theory, enabling the construction of new graphs with specific structures. They have significant applications in various fields, including computer science, network analysis, chemistry, and physics, supporting both theoretical research and practical use. Edge corona of graphs is one such operation. Let [Formula: see text] be a graph with [Formula: see text] vertices and [Formula: see text] edges. The edge corona [Formula: see text] of two graphs [Formula: see text] and [Formula: see text] is obtained by taking one copy of [Formula: see text] and [Formula: see text] copies of [Formula: see text] and making the end vertices of the [Formula: see text] edge of [Formula: see text] adjacent to all the vertices in the [Formula: see text] copy of [Formula: see text], where [Formula: see text]. Let [Formula: see text] be the family of graphs that includes complete graphs with at least [Formula: see text] vertices and graphs of diameter [Formula: see text], in which the diametrically opposite vertices of any peripheral vertex are non-adjacent. In this paper, we describe the reciprocal distance spectrum of [Formula: see text] in terms of the adjacency spectra of [Formula: see text] and [Formula: see text] when both [Formula: see text] and [Formula: see text] are regular and [Formula: see text]. Several constructions are proposed using line graphs, iterated line graphs, double graphs, strong double graphs and complement graphs to obtain infinitely many reciprocal distance non-cospectral pairs of reciprocal distance equienergetic graphs and non-isomorphic pairs of reciprocal distance cospectral graphs.