Some spectral invariants of the neighborhood corona of graphs

Given two graphs, G1 with vertices {v1,v2,…,vn} and G2, the neighborhood corona, G1⋆G2, is the graph obtained by taking n copies of G2 and joining by an edge each neighbor of vi, in G1, to every vertex of the ith copy of G2. A special instance G1⋆K1 of the neighborhood corona is called the splitting graph of G1 and has a property that its spectrum consists of all eigenvalues ϕλ and −ϕ−1λ, where ϕ=(1+5)∕2 is the golden ratio and λ is an arbitrary eigenvalue of G1. In this paper, various spectra invariants of the neighborhood corona of graphs are studied. First, the condition number, the inertia, and the HOMO-LUMO gap of the s-fold splitting graphs are investigated, some of which turn out to have the golden-ratio scaling with the corresponding invariants of the original graph. Then, resistance distances and the Kirchhoff index of the neighborhood corona graph G1⋆G2 are computed, with explicit expressions being obtained, which extends the previously known result.