On Some Seidel Integral Graphs

The set of all the eigenvalues of a graph is called its spectrum. Let G be a simple graph of order n. Its Seidel adjacency matrix is denoted by A^*(G) = J - I - 2A. A graph G is called Seidel integral if its Seidel spectrum consists entirely of integers. The estimation on graphs by its spectrum is an active question for discussion, moreover integral graph is a special class. There are many results about integral graphs. But for Seidel integral graphs, there are few about it. This paper determines all the Seidel integral graphs ofαK_a∪βK_b,αK_a∪βCP(b) andαK_a,a∪βCP(b) mainly by the main eigenvalue relation between adjacency matrix and Seidel adjacency matrix and Diophantine equation. In this paper, we obtain thatDetermination about all the Seidel integral graphs ofαK_a∪βK_b,αK_a∪βCP(b) andαK_a,a∪βCP(b). For example:LetαK_a∪βK_b is seidel integral, then its form is followed:among (i) t, k,l, n,m,f,e, h∈N, and (m,n) = 1, (f, e) = 1;(iv) (?), andÏ„| kt;(iii) (x₀, y₀) is a special root of (?);(iv) z≥z₀, z₀ is minimal integral number which satisfy the following condition,Finally we make apply of the main eigenvalue relation and Diophantine equation, and obtain the detail forms. That is the main part of this paper.