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)) ¯ = { x y ≠ E (I (ℤ)) : x , y ∈ V (I (ℤ)) }. In this paper we determine the Laplacian spectrum of the complement of identity graph I (ℤ 2 p) ¯ , for some prime p, that can be constructed by investigating the eigenvalues of I (ℤ 2 p) ¯. The result shows that all eigenvalues of I (ℤ 2 p) ¯ are integers. [ABSTRACT FROM AUTHOR]