Hein van der Holst

Graphs whose minimal rank is two

Let F be a field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F, G )co nsists of the symmetric irreducible tridiagonal matrices. Let mr(F, G) be the minimum rank over all matrices in S(F, G). Then mr(F, G) = 1 if and only if G is the union of a clique with at least 2 vertices and an independent set. If F is an infinite field such that charF � , then mr(F, G) ≤ 2i f and only if the complement of G is the join of a clique and a graph that is the union of at most two cliques and any number of complete bipartite graphs. A similar result is obtained in the case that F is an infinite field with char F = 2. Furthermore, in each case, such graphs are characterized as those for which 6 specific graphs do not occur as induced subgraphs. The number of forbidden subgraphs is reduced to 4 if the graph is connected. Finally, similar criteria is obtained for the minimum rank of a Hermitian matrix to be less than or equal to two. The complement is the join of a clique and a graph that is the union of any number of cliques and any number of complete bipartite graphs. The number of forbidden subgraphs is now 5, or in the connected case, 3.

ON THE MINIMUM RANK OF NOT NECESSARILY SYMMETRIC MATRICES: A PRELIMINARY STUDY ∗

The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i,j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for ij) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.

Graphs whose minimal rank is two : the finite fields case

Let F be a finite field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G.L et mr(F, G) be the minimum rank of all matrices in S(F, G). If F is a finite field with p t elements, p � , it is shown that mr(F, G) ≤ 2i f and only if the complement of G is the join of a complete graph with either the union of at most (p t +1)/2 nonempty complete bipartite graphs or the union of at most two nonempty complete graphs and of at most (p t − 1)/2 nonempty complete bipartite graphs. These graphs are also characterized as those for which 9 specific graphs do not occur as induced subgraphs. If F is a finite field with 2t elements, then mr(F, G) ≤ 2 if and only if the complement of G is the join of a complete graph with either the union of at most 2t + 1 nonempty complete graphs or the union of at most one nonempty complete graph and of at most 2 t−1 nonempty complete bipartite graphs. A list of subgraphs that do not occur as induced subgraphs is provided for this case as well.

ON THE MAXIMUM POSITIVE SEMI-DEFINITE NULLITY AND THE CYCLE MATROID OF GRAPHS ∗

Let G =( V,E )b e ag raph withV = {1,2,...,n}, in which we allow parallel edges but no loops, and let S+(G) be the set of all positive semi-definite n × n matrices A =( ai,j )w ith ai,j =0i fij and i and j are non-adjacent, ai,jfij and i and j are connected by exactly one edge, and ai,j ∈ R if i = j or i and j are connected by parallel edges. The maximum positive semi-definite nullity of G, denoted by M+(G), is the maximum nullity attained by any matrix A ∈S +(G). A k-separation of G is a pair of subgraphs (G1,G2) such that V (G1) ∪ V (G2 )= V , E(G1) ∪ E(G2 )= E, E(G1) ∩ E(G2 )= ∅ and |V (G1) ∩ V (G2)| = k .W henG has a k-separation (G1,G2 )w ithk ≤ 2, we give a formula for the maximum positive semi-definite nullity of G in terms of G1,G2 ,a nd in case ofk = 2, also two other specified graphs. For a graph G ,l etcG denote the number of components in G. As a corollary of the result on k-separations with k ≤ 2, we obtain that M+(G) − cG = M+(G � ) − cG for graphs G and Gthat have isomorphic cycle matroids.

Sign patterns with minimum rank 2 and upper bounds on minimum ranks

A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0}. The minimum rank (resp., rational minimum rank) of a sign pattern matrix 𝒜 is the minimum of the ranks of the real (resp., rational) matrices whose entries have signs equal to the corresponding entries of 𝒜. The notion of a condensed sign pattern is introduced. A new, insightful proof of the rational realizability of the minimum rank of a sign pattern with minimum rank 2 is obtained. Several characterizations of sign patterns with minimum rank 2 are established, along with linear upper bounds for the absolute values of an integer matrix achieving the minimum rank 2. A known upper bound for the minimum rank of a (+, −) sign pattern in terms of the maximum number of sign changes in the rows of the sign pattern is substantially extended to obtain upper bounds for the rational minimum ranks of general sign pattern matrices. The new concept of the number of polynomial sign changes of a sign vector is crucial for this extension. Another known upper bound for the minimum rank of a (+, −) sign pattern in terms of the smallest number of sign changes in the rows of the sign pattern is also extended to all sign patterns using the notion of the number of strict sign changes. Some examples and open problems are also presented.

A polynomial-time algorithm to find a linkless embedding of a graph

A Z-linkless embedding of a graph is an embedding in 3-space such that each pair of disjoint circuits has zero linking number. In this paper we present polynomial-time algorithms to compute a Z-linkless embedding of a graph provided the graph has one and to test whether an embedding of a graph is Z-linkless or not.

Algebraic characterizations of outerplanar and planar graphs

A drawing of a graph in the plane is even if nonadjacent edges have an even number of intersections. Hananis theorem characterizes planar graphs as those graphs that have an even drawing. In this paper we present an algebraic characterization of graphs that have an even drawing. Together with Hananis theorem this yields an algebraic characterization of planar graphs. We will also present algebraic characterizations of subgraphs of paths, and of outerplanar graphs.

Graphs and obstructions in four dimensions

For any graph G = (V, E) without loops, let C2(G) denote the regular CW-complex obtained from G by attaching to each circuit C of G a disc. We show that if G is the suspension of a flat graph, then C2(G) has an embedding into 4-space. Furthermore, we show that for any graph G in the collection of graphs that can be obtained from K7 and K3,3,1,1 by a series of ΔY- and YΔ-transformations, c2(G) cannot be embedded into 4-space.

Two Tree-Width-Like Graph Invariants

In this paper we introduce two tree-width-like graph invariants. The first graph invariant, which we denote by ν=(G), is defined in terms of positive semi-definite matrices and is similar to the graph invariant ν(G), introduced by Colin de Verdière in [J. Comb. Theory, Ser. B., 74:121–146, 1998]. The second graph invariant, which we denote by θ(G), is defined in terms of a certain connected subgraph property and is similar to λ(G), introduced by van der Holst, Laurent, and Schrijver in [J. Comb. Theory, Ser. B., 65:291–304, 1995]. We give some theorems on the behaviour of these invariants under certain transformations. We show that ν=(G)=θ(G) for any graph G with ν=(G)≤4, and we give minimal forbidden minor characterizations for the graphs satisfying ν=(G)≤k for k=1,2,3,4.

On a graph property generalizing planarity and flatness

We introduce a topological graph parameter σ(G), defined for any graph G. This parameter characterizes subgraphs of paths, outerplanar graphs, planar graphs, and graphs that have a flat embedding as those graphs G with σ(G)≤1,2,3, and 4, respectively. Among several other theorems, we show that if H is a minor of G, then σ(H)≤σ(G), that σ(Kn)=n−1, and that if H is the suspension of G, then σ(H)=σ(G)+1. Furthermore, we show that µ(G)≤σ(G) + 2 for each graph G. Here µ(G) is the graph parameter introduced by Colin de Verdière in [2].