Abstract:It was proved (A. Kotlov and L. Lovász, The rank and size of graphs, J. Graph Theory 23 (1996), 185-189) that the number of vertices in a twin-free graph is O(( √ 2) r ) where r is the rank of the adjacency matrix. This bound was shown to be tight. We show that the chromatic number of a graph is o(∆ r ) where ∆ = 4/3 < √ 2.

ANDREȊ KOTLOVLet G H and G2H denote, respectively, the strong and Cartesian products of graphs G and H . (We recall that K 2 K 2 is the complete graph K 4 on four vertices, while K 2 2K 2 is a four-cycle C 4 .) Using a simple construction, we show that, for every bipartite G, the graph G K 2 is a minor of the graph G2C 4 . In particular, the d-cube Q d has a complete minor on 2 (d+1)/2 vertices if d is odd, and on 3 · 2 (d−2)/2 vertices if d is even. We do not know whether such a complete minor of Q d is largest possible.

Abstract. Let F be a field. Given a simple graph G on n vertices, its minimal rank (with respect to F ) is the minimum rank of a symmetric n × n F -valued matrix whose off-diagonal zeroes are the same as in the adjacency matrix of G. If F is finite, then for every k, it is shown that the set of graphs of minimal rank at most k is characterized by finitely many forbidden induced subgraphs, each on at most ( |F | k 2 + 1) 2 vertices. These findings also hold in a more general context.

In his recent preprint "Multiplicities of eigenvalues and tree-width of graphs," Colin de Verdière introduced a new spectral invariant (denoted here by (G)) of a graph G, similar in spirit to his now-classical invariant µ(G). He showed that (G) is minor-monotone and is related to the tree-width la(G) of G: (G) ≤ la(G) and, moreover, (G) ≤ 1 ⇔ la (G) = 1, i.e. G is a forest. We show that (G) = 2 ⇔ la (G) = 2 and give the corresponding forbidden-minor and ear-decomposition characterizations.

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