Some Results On The Edge Corona

Given two simple graphs G₁ and G₂?G₁has m edges?,the edge corona of G₁ and G₂is defined as the graph obtained by taking m copies of G₂,denoted by G₂¹,G₂²,…,G₂^m,and for each edge ei=?u,v?of G₁,joining edges between each of two end-vertices u and v of eiand each vertex of Gk₂,which is denoted by G₁? G₂.For a simple graph G,we do a graph operation of G to R?G?,which is obtained by changing every edge of G to be a triangle.This is equivalent to the edge corona of G and K₁,i.e.R?G?= G ? K₁.We obtain R²?G?by doing graph operation R?G?? K₁,and we obtain R³?G?by doing graph operation of R²?G?? K₁,and so on.Hence Rⁿ?G?= R^n-1?G?? K₁,n = 2,3,….Zhang et al.?Physica A,391?2012?,828-833?studied the Monomer-dimer problem of Rⁿ?K₃?.They obtained the exact formula for the number of all possible monomer-dimer arrangements and the asymptotic growth constant on Rⁿ?K₃?.In the second chapter of this paper,we generalize this result and obtain the exact solution on the monomer-dimer model on Rⁿ?G?.Particularly,we prove that for an arbitrary graph G,Rⁿ?G?has the same asymptotic growth constant of the number of monomer-dimer arrangements.The spectra of the adjacency,Laplacian,signless Laplacian and normalized Laplacian matrices of the edge corona graph G₁? G₂,have been studied relatively perfect.But the spectra of the modified edge corona graph G₁[G₂] have not been studied.In the third chapter of this paper we solve these problems.We also give the exact formulae of the number of spanning trees and the Kirchhoff index of G₁[G₂].