布尔矩阵的可实现问题及其与色数问题的关系

讨论了布尔矩阵的可实现问题及其与色数问题的关系.首先给出布尔矩阵可实现的一些充要条件,讨论可实现布尔矩阵的性质,其次证明可实现布尔矩阵的容度等于该矩阵所生成的图的色数;简单图的邻接矩阵的对偶阵是可实现的,且其容度就是简单图的色数的一个上界.     

Upper bounds on the independence and the clique covering number

New upper bounds for the independence number and for the clique covering number of a graph are given in terms of the rank, respectively the eigenvalues, of the adjacency matrix. We formulate a conjecture concerning an upper bound of the clique covering number. This upper bound is related to an old conjecture of Alan J. Hoffman which is shown to be false. Key words: adjacency matrix, eigenvalues, independence number, clique covering number. AMS classification: 05C.     

A characterization of graphs with rank 4

The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: What is the structure of a connected graph with rank 4? This question has not yet been fully answered in the literature, and only some partial results are known. In this paper we resolve this question by completely characterizing graphs G whose adjacency matrix has rank 4.     

On the “log rank”-conjecture in communication complexity

We show the existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix. We consider the following problem: each of two players gets a perfect matching between twon-element sets of vertices. Their goal is to decide whether or not the union of the two matcliings forms a Hamiltonian cycle. We prove:

1. The rank of the input matrix over the reals for this problem is 2 ^O(n) .
2. The non-deterministic communication complexity of the problem is Ω(nloglogn).

Our result also supplies a superpolynomial gap between the chromatic number of a graph and the rank of its adjacency matrix. Another conclusion from the second result is an Ω(nloglogn) lower bound for the graph connectivity problem in the non-deterministic case. We make use of the theory of group representations for the first result. The second result is proved by an information theoretic argument.     

秩$6$的无三角形的图

The rank of a graph G is defined to be the rank of its adjacency matrix A（G）. In this paper we characterize all connected triangle-free graphs with rank 6.     

整循环图的秩（英文）

A graph is called an integral graph if it has an integral spectrum i.e.,all eigenvalues are integers.A graph is called circulant graph if it is Cayley graph on the circulant group,i.e.,its adjacency matrix is circulant.The rank of a graph is defined to be the rank of its adjacency matrix.This importance of the rank,due to applications in physics,chemistry and combinatorics.In this paper,using Ramanujan sums,we study the rank of integral circulant graphs and gave some simple computational formulas for the rank and provide an example which shows the formula is sharp.     

The rank of random graphs

We show that almost surely the rank of the adjacency matrix of the Erd?s‐Rényi random graph G(n,p) equals the number of nonisolated vertices for any c ln n/n ≤ p ≤ 1/2, where c is an arbitrary positive constant larger than 1/2. In particular, the adjacency matrix of the giant component (a.s.) has full rank in this range. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Fuzzy矩阵Schein秩的计算复杂性

本文讨论Fuzzy矩阵Schein秩的计算复杂性问题,证明了它是一个"NP-完全问题".首先,刻画了交可分解的Puzzy关系的交分解解集.然后,从Fuzzy关系的交分解与广义分解之间的关系出发,给出了Fuzzy关系广义分解的算法.最后,从Fuzzy关系广义分解的角度来讨论Fuzzy矩阵的Schein秩.指出它与色数问题之间的关系,即Fuzzy矩阵的Schein秩等于由它生成的简单图的色数,从而证明了计算Fuzzy矩阵的Schein秩是一个"NP-完全问题".     

Maximum Order of Triangle‐Free Graphs with a Given Rank

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced triangle‐free graphs with a given rank and characterize all such graphs achieving the maximum order.     

The rank and size of graphs

We show that the number of points with pairwise different sets of neighbors in a graph is O(2^r/2), where r is the rank of the adjacency matrix. We also give an example that achieves this bound. © 1996 John Wiley & Sons, Inc.     

Orthogonal representations over finite fields and the chromatic number of graphs

We study the relationship between the minimum dimension of an orthogonal representation of a graph over a finite field and the chromatic number of its complement. It turns out that for some classes of matrices defined by a graph the 3-colorability problem is equivalent to deciding whether the class defined by the graph contains a matrix of rank 3 or not. This implies the NP-hardness of determining the minimum rank of a matrix in such a class. Finally we give for any class of matrices defined by a graph that is interesting in this respect a reduction of the 3-colorability problem to the problem of deciding whether or not this class contains a matrix of rank equal to three.The author is financially supported by the Cooperation Centre Tilburg and Eindhoven Universities.     

Decomposable polymatroids and connections with graph coloring

We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of rank functions of matroids, and for which the minimum number of matroids required in such sums is the chromatic number of the line graph of the hypergraph. This result motivates introducing chromatic numbers and chromatic polynomials for polymatroids. We show that the chromatic polynomial of any 2-polymatroid is a rational multiple of the chromatic polynomial of some graph. We also find the excluded minors for the minor-closed class of polymatroids that can be written as sums of rank functions of matroids that form a chain of quotients.     

图同构的必要条件

结合图对应的邻接矩阵,利用矩阵的秩和矩阵的合同关系,得到了图同构的一个必要条件;然后给出了图同构的一个理论判断的算法．     

Maximum order of trees and bipartite graphs with a given rank

The rank of a graph is that of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced trees as well as bipartite graphs with a given rank and characterize those graphs achieving the maximum order.     

Spectral bounds for the independence ratio and the chromatic number of an operator

We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L ²-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical framework in which many packing and coloring problems for finite and infinite graphs can be conveniently studied with the help of harmonic analysis and convex optimization. The theory is applied to infinite geometric graphs on Euclidean space and on the unit sphere.     

On the length of longest chordless cycles

Upper bounds for the length of a longest (circuit) cycle without chords in a (directed) graph are given in terms of the rank of the adjacency matrix and in terms of its eigenvalues.Received: October 2003, Revised: January 2005, AMS classification: 05C     

Some relations among term rank, clique number and list chromatic number of a graph

Let G be a graph with a nonempty edge set, we denote the rank of the adjacency matrix of G and term rank of G, by rk(G) and Rk(G), respectively. van Nuffelen conjectured that for any graph G, χ(G)?rk(G). The first counterexample to this conjecture was obtained by Alon and Seymour. In 2002, Fishkind and Kotlov proved that for any graph G, χ(G)?Rk(G). Here we improve this upper bound and show that χ_l(G)?(rk(G)+Rk(G))/2, where χ_l(G) is the list chromatic number of G.     

On the adjacency matrix of a threshold graph

A threshold graph on n   vertices is coded by a binary string of length n−1$n-1$. We obtain a formula for the inertia of (the adjacency matrix of) a threshold graph in terms of the code of the graph. It is shown that the number of negative eigenvalues of the adjacency matrix of a threshold graph is the number of ones in the code, whereas the nullity is given by the number of zeros in the code that are preceded by either a zero or a blank. A formula for the determinant of the adjacency matrix of a generalized threshold graph and the inverse, when it exists, of the adjacency matrix of a threshold graph are obtained. Results for antiregular graphs follow as special cases.     

The chromatic number and rank of the complements of the Kasami graphs

We determine the rank and chromatic number of the complements of all Kasami graphs, some of which form an infinite family of counterexamples to the now disproven rank-coloring conjecture.