On the Seidel integral graphs which belong to the class αKa,a ∪ βKb,b

Abstract

We say that a simple graph G is Seidel integral if its Seidel spectrum consists entirely of integers. If ?Ka,a ? ?Kb,b is Seidel integral, we show that it belongs the class of Seidel integral graphs [kt/? x0 + mt/?z]Ka,a ? [kt/? y0 + a/?z]nKb,b, where (i) a = (t+?n)k+?m and b = ?m; (ii) t, k, ?,m, n ? N such that (m, n) = 1, (n, t) = 1 and (?, t) = 1; (iii) ? = (a,mt) such that ? | kt; (iv) (x0, y0) is a particular solution of the linear Diophantine equation ax ? (mt)y = ? and (v) z ? z0 where z0 is the least integer such that (kt/? x0 + mt/?z0)? 1 and (kt/? y0 + a/?z0) ? 1. In particular, we demonstrate that ?Ka ? ?Kb is integral in respect to its ordinary adjacency matrix if and only if ?Ka,a ? ?Kb,b is Seidel integral.