Some spectral invariants of the neighborhood corona of graphs

Given two graphs, G(1) with vertices {v(1), v(2),., v(n)} and G(2), the neighborhood corona, G(1) * G(2), is the graph obtained by taking n copies of G(2) and joining by an edge each neighbor of v(i), in G(1), to every vertex of the ith copy of G(2). A special instance G(1) * K-1 of the neighborhood corona is called the splitting graph of G1 and has a property that its spectrum consists of all eigenvalues phi lambda and -phi(-1)lambda, where phi = (1 +root 5)/2 is the golden ratio and lambda is an arbitrary eigenvalue of G(1). 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 G(1) * G(2) are computed, with explicit expressions being obtained, which extends the previously known result. (C) 2018 Elsevier B.V. All rights reserved.