New class of integral bipartite graphs with large diameter

Document Type: Research Paper


Shahed university, Tehran, Iran.


In this paper‎, ‎we construct a new class of integral bipartite graphs (not necessarily trees) with large even diameters‎. ‎In fact‎, ‎for every finite set $A$ of positive integers of size $k$ we construct an integral bipartite graph $G$ of diameter $2k$ such that the set of positive eigenvalues of $G$ is exactly $A$‎. ‎This class of integral bipartite graphs has never found before‎.


Main Subjects

[1] A. E. Brouwer and W. H. Haemers, Spectra of graphs, Universitext, Springer, New York, 2012.
[2] P. Csikvári, Integral trees of arbitrarily large diameters, J. Algebraic Combin., 32 (2010) 371–377.
[3] D. Cvetkovi´c, P. Rowlinson and S. Simi´c, An Introduction to the Theory of Graph Spectra, 75, Cambridge University Press, Cambridge, 2010.

[4] E. Ghorbani, A. Mohammadian and B. Tayfeh-Rezaie, Integral trees of odd diameters, J. Graph Theory, 70 (2012) 332–338.
[5] F. Harary and A. J. Schwenk, Which graphs have integral spectra?, Graphs and Combinatorics, Lecture Notes in Math., 406, Springer, Berlin, (1974) 45–51.
[6] M. Watanabe and A. J. Schwenk, Integral starlike trees, J. Austral. Math. Soc. Ser. A, 28 (1979) 120–128.