vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i ̸ = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues.

We show that the Steinitz representations of 3-connected planar graphs are correspond, in a well described way, to Colin de Verdiere matrices of such graphs. 1 Introduction Colin de Verdiere [1] introduced a spectral invariant (G) of a graph G. Roughly speaking, (G) is the multiplicity of the second largest eigenvalue of the adjacency matrix of G, maximized by weighting the edges and nodes (the exact definition will be given in section 3). The matrix that attains this maximum will be called a Colin de Verdiere matrix of G. This spectral invariant has many interesting unexpected properties. The starting point of this paper are the facts, proved by Colin de Verdiere, that a graph G is outerplanar if and only if (G) # 2, and planar if and only if (G) # 3. Suppose that G is a 3-connected planar graph, and consider its Colin de Verdiere matrix M . Then the appropriate eigenvalue of M has multiplicity 3. It was shown in [7] that one can use the corresponding eigensubspace to construc...

In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless embeddability of G is characterized by the inequality (G) 4. In this paper we give an overview of results on (G) and of techniques to handle it.

In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete n-dimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.

The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, construction methods for geometric representations, and their applications in proofs and algorithms.

A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3-space, Hadwiger’s conjecture on t-colorability of graphs with no Kt+1 minor, Tutte’s edge 3-coloring conjecture on edge 3-colorability of 2-connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2-colorability of hypergraphs, and sign-nonsingular matrices.

We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3-coloring conjecture, and Pfaffian orientations of bipartite graphs.

Abstract not found

. Let G = (V; E) be a 3-connected planar graph, with V = f1; : : : ; ng. Let M = (m i;j ) be a symmetric n \Theta n matrix with exactly one negative eigenvalue (of multiplicity 1), such that for i; j with i 6= j, if i and j are adjacent then m i;j ! 0 and if i and j are nonadjacent then m i;j = 0, and such that M has rank n \Gamma 3. Then the null space ker M of M gives an embedding of G in S 2 as follows: Let a; b; c be a basis of ker M , and for i 2 V let OE(i) := (a i ; b i ; c i ) T ; then OE(i) 6= 0, and /(i) := OE(i)=kOE(i)k embeds V in S 2 such that connecting, for any two adjacent vertices i; j, the points /(i) and /(j) by a shortest geodesic on S 2 , gives a proper embedding of G in S 2 . This applies to the matrices associated with the parameter (G) introduced by Y. Colin de Verdi`ere. 1 Introduction and results Let G = (V; E) be an undirected graph, with vertex set f1; : : : ; ng. Let M(G) be the set of symmetric n \Theta n matrices M = (m i;j ) satisfying (i) M...