THE LAPLACIAN SPECTRUM OF GRAPHS † Bojan Mohar ‡ Department of Mathematics University of Ljubljana Jadranska 19, 61111 Ljubljana Yugoslavia Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Lapla- cian eigenvalue λ 2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2, Ed. Y. Alavi, G. Chartrand, O. R. Oellermann, A. J. Schwenk, Wiley, 1991, pp. 871–898. ‡ The work supported in part by the Research Council of Slovenia, Yugoslavia. Part of the work was done while the author was a Fulbright Scholar at the Ohio State University, Columbus, Ohio. 1