The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. Among nvertex trees, the star has greatest nullity (equal to n − 2). We generalize this by showing that among n-vertex trees whose vertex degrees do not exceed a certain value D, the greatest nullity is n − 2 (n − 1)/D . Methods for constructing such maximum-nullity trees are described.

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a b s t r a c tA function between graphs is k-to-1 if each point in the co-domain has precisely k preimages in the domain. Given two graphs, G and H, and an integer k ≥ 1, and considering G and H as subsets of R 3 , there may or may not be a k-to-1 continuous function (i.e. a k-to-1 map in the usual topological sense) from G onto H. In this paper we review and complete the determination of whether there are finitely discontinuous, or just infinitely discontinuous k-to-1 functions between two intervals, each of which is one of the following: ]0, 1[, [0, 1[ and [0, 1]. We also show that for k even and 1 ≤ r < 2s, (r, s) = (1, 1) and (r, s) = (3, 2), there is a k-to-1 map from K 2r onto K 2s if and only if k ≥ 2s.

In papers (7, 8), Vizing studied graphs which are critical with respect to edge colourings. In particular, he obtained bounds on the number of edges and on the circuit length of such graphs in terms of its maximum valency. The object of this paper is to improve these bounds by obtaining others which also take into account the order of the graph.