Abstract
The Kirchhoff index ofGis the sum of resistance distances between all pairs of vertices ofGin electrical networks.LEL(G)is the Laplacian-Energy-Like Invariant ofGin chemistry. In this paper, we define two classes of join graphs: the subdivision-vertex-vertex joinG1⊚G2and the subdivision-edge-edge joinG1⊝G2. We determine the generalized characteristic polynomial of them. We deduce the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomials ofG1⊚G2andG1⊝G2whenG1isr1-regular graph andG2isr2-regular graph. As applications, the Laplacian spectra enable us to get the formulas of the number of spanning trees, Kirchhoff index, andLELofG1⊚G2andG1⊝G2in terms of the Laplacian spectra ofG1andG2.
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