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 λ2 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,

this article we suggest a family of weights a k , derived from the irreducible characters of the symmetric group, that would include (1) and (2) as special cases. This generalization provides much insight into the nature of the identities (1) and (2), and enables us to appreciate some of the properties such as why the Email: matlamtk@math.nus.sg L(G) = B B B B B B B B B B B \Gamma1 1 0 \Gamma1 1 . .