On integral graphs which belong to the class$\overline {\alpha K_a \cup \beta K_b }$

Abstract

LetG be a simple graph and let$\bar G$ denotes its complement. We say thatG is integral if its spectrum consists entirely of integers. If$\overline {\alpha K_a \cup \beta K_b }$ is integral we show that it belongs to the class of integral graphs

$$\overline {[\frac{{kt}}{\tau }x_o + \frac{{mt}}{\tau }z]K(t + \ell n)k + \ell m \cup [\frac{{kt}}{\tau }y_o + \frac{{(t + \ell n)k + \ell m}}{\tau }z]nK\ell m,} $$

where (i) t, k, l, m, n ∈ ℕ such that (m, n) =1, (n, t) =1 and (l, t)=1; (ii) τ=((t+ln)k+lm, mt) such that τ| kt; (iii) (x₀, y₀) is aparticular solution of the linear Diophantine equation ((t+ln)k+lm)x-(mt)y=τ and (iv) z≥z₀ where z₀ is the least integer such that$(\frac{{kt}}{\tau }x_0 + \frac{{mt}}{\tau }z_0 ) \geqslant 1$ and$(\frac{{kt}}{\tau }y_0 + \frac{{(t + \ell n)k + \ell m}}{\tau }z_0 ) \geqslant 1$.