Laplacian spectrum of the complement of identity graph of commutative ring ℤ2p

Research on the algebraic graph theory is still being developed by many researchers. Let ℤ be a commutative ring. A graph I(ℤ) is a graph with a vertex set of units ℤ and x, y∈ℤ, x≠y, are adjacent if and only if x. y=1, and all vertices adjacent to 1. The complement of I(ℤ)= (V(I(ℤ)), E(I(ℤ))), denoted by I(ℤ)¯=(V(I(ℤ))¯,E(I(ℤ¯))), is a graph with V(I(ℤ))¯=V(I(ℤ)) and E(I(ℤ))¯={xy≠E(I(ℤ)):x,y∈V(I(ℤ))}. In this paper we determine the Laplacian spectrum of the complement of identity graph I(ℤ2p)¯, for some prime p, that can be constructed by investigating the eigenvalues of I(ℤ2p)¯. The result shows that all eigenvalues of I(ℤ2p)¯ are integers.