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
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\((\frac{{kt}}{\tau }y_0 + \frac{{(t + \ell n)k + \ell m}}{\tau }z_0 ) \geqslant 1\).
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References
D. Cvetković, M. Doob and H. Sachs,Spectra of graphs — Theory and applications, 3rd revised and enlarged edition, J. A. Barth Verlag, Heidelberg, — Leipzig, 1995.
G. H. Hardy and E. M. Wright,An introduction to the theory of numbers, 4th edition, Oxford University Press, 1960.
M. Lepović,Some results on graphs with exactly two main eigenvalues, Univ. Beograd Publ. Elektro-tehn. Fak. (Ser. Mat.)12 (2001), 68–84.
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Mirko Lepović defended his doctoral theses 1991, under the name “Solving some hereditary problems in the Spectral theory of graphs”, at Belgrade University. Since 1993 he has worked at University of Kragujevac (Department of Mathematics). His main areas of research are in graph theory, linear algebra and combinatorics. He has also interested on computer programming.
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Lepović, M. On integral graphs which belong to the class\(\overline {\alpha K_a \cup \beta K_b } \) . JAMC 14, 39–49 (2004). https://doi.org/10.1007/BF02936097
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DOI: https://doi.org/10.1007/BF02936097