Generalized Characteristic Polynomials of Join Graphs and Their Applications

Abstract EN

The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices of G in electrical networks.

LEL(G) is the Laplacian-Energy-Like Invariant of G in chemistry.

In this paper, we define two classes of join graphs: the subdivision-vertex-vertex join G1⊚G2 and the subdivision-edge-edge join G1⊝G2.

We determine the generalized characteristic polynomial of them.

We deduce the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomials of G1⊚G2 and G1⊝G2 when G1 is r1-regular graph and G2 is r2-regular graph.

As applications, the Laplacian spectra enable us to get the formulas of the number of spanning trees, Kirchhoff index, and LEL of G1⊚G2 and G1⊝G2 in terms of the Laplacian spectra of G1 and G2.