Diffusion determines the recurrent graph

We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown by counterexamples this result is optimal. Without the recurrence assumption, the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each point has the same mass.     

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.     

双线性型图与交错型图不是完美图

本文证明了双线性型图与交错型图都不是完美图,从而解决了双线性型图与交错型图的完美图判别问题.     

The Totally Non-positive Matrix Completion Problem

In this paper, the totally non-positive matrix is introduced. The totally non-positive completion asks which partial totally non-positive matrices have a completion to a totally non-positive matrix. This problem has. in general, a negative answer. Therefore, our question is for what kind of labeled graphs G each partial totally non-positive matrix whose associated graph is G has a totally non-positive completion? If G is not a monotonically labeled graph or monotonically labeled cycle, we give necessary and sufficient conditions that guarantee the existence of the desired completion.     

Inverting graphs of rectangular matrices

This paper addresses the question of determining the class of rectangular matrices having a given graph as a row or column graph. We also determine equivalent conditions on a given pair of graphs in order for them to be the row and column graphs of some rectangular matrix. In connection with these graph inversion problems we discuss the concept of minimal inverses. This concept turns out to have two different forms in the case of one-graph inversion. For the two-graph case we present a method of determining when an inverse is minimal. Finally we apply the two-graph theorem to a class of energy related matrices.     

On Balanced Graphs

Berge defined a hypergraph to be balanced if its incidence matrix is balanced. We consider this concept applied to graphs, and call a graph to be balanced when its clique matrix is balanced. Characterizations of balanced graphs by forbidden subgraphs and by clique subgraphs are proved in this work. Using properties of domination we define four subclasses of balanced graphs. Two of them are characterized by 0–1 matrices and can be recognized in polynomial time. Furthermore, we propose polynomial time combinatorial algorithms for the problems of stable set, clique-independent set and clique-transversal for one of these subclasses of balanced graphs. Finally, we analyse the behavior of balanced graphs and these four subclasses under the clique graph operator. Received: April, 2004     

线团-收敛图

一个图的线团图就是这个图的线图的团图。对于自然数n，一个图被称为n-线团－收敛的，如果它的n次线团图同构于一个固定的图。否则称之为发散的。本刻画了线团－收敛图与发散图，给出一个线团－收敛图的构造方法，并且，讨论了线团－收敛图的线团－收敛指数。     

On graphs whose Laplacian matrix’s multipartite separability is invariant under graph isomorphism

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are responsible for many nonintuitive physical phenomena. The entanglement property of normalized Laplacian matrices is in general not invariant under graph isomorphism. In recent papers, graphs were identified whose entanglement and separability properties are invariant under isomorphism. The purpose of this note is to completely characterize the set of graphs whose separability is invariant under graph isomorphism. In particular, we show that this set consists of K_2,2 and its complement, all complete graphs and no other graphs.     

Self-clique graphs and matrix permutations

The clique graph of a graph is the intersection graph of its (maximal) cliques. A graph is self-clique when it is isomorphic with its clique graph, and is clique-Helly when its cliques satisfy the Helly property. We prove that a graph is clique-Helly and self-clique if and only if it admits a quasi-symmetric clique matrix, that is, a clique matrix whose families of row and column vectors are identical. We also give a characterization of such graphs in terms of vertex-clique duality. We describe new classes of self-clique and 2-self-clique graphs. Further, we consider some problems on permuted matrices (matrices obtained by permuting the rows and/or columns of a given matrix). We prove that deciding whether a (0,1)-matrix admits a symmetric (quasi-symmetric) permuted matrix is graph (hypergraph) isomorphism complete. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 178–192, 2003     

Weight choosability of graphs

Suppose the edges of a graph G are assigned 3‐element lists of real weights. Is it possible to choose a weight for each edge from its list so that the sums of weights around adjacent vertices were different? We prove that the answer is positive for several classes of graphs, including complete graphs, complete bipartite graphs, and trees (except K₂). The argument is algebraic and uses permanents of matrices and Combinatorial Nullstellensatz. We also consider a directed version of the problem. We prove by an elementary argument that for digraphs the answer to the above question is positive even with lists of size two. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 242–256, 2009     

Algebraic models for two-edge-connected graphs

The purpose of this paper is to associate two-edge-connected graphs to a vector space of matrices. The fruitfulness of this association is shown by deriving various graph theoretic results via algebraic arguments on the associated vector space.The paper is divided into two sections. In the first section, an association between a vector space of matrices and two-edge-connected graphs is developed. The transfer of various properties of graphs to matrices and of matrices to graphs is then considered. In the second section the utility of the association developed in Section 1 is indicated by showing how the algebraic representation may be used, as a tool, to deduce rather interesting results concerning the structure of the associated graphs.     

Arc graphs and their possible application to sparse matrix problems

In continuation of earlier work on the graph algorithmic language GRAAL, a new type of graph representation is introduced involving solely the arcs and their incidence relations. In line with the set theoretical foundation of GRAAL, the are graph structure is defined in terms of four Boolean mappings over the power set of the ares. A simple data structure is available for are graphs requiring only storage of the order of the cardinality of the are set. As an application, the LU decomposition of large sparse matrices and the solution of the corresponding linear systems are formulated in terms of are graphs and their operators, and experimental results involving these algorithms are presented.     

The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States

We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss the problem of separability by pointing out that separability of density matrices of graphs does not always depend on the labelling of the vertices. We consider graphs with a tensor product structure and simple cases for which combinatorial properties are linked to the entanglement of the state. We calculate the concurrence of all graphs on four vertices representing entangled states. It turns out that for these graphs the value of the concurrence is exactly fractional. Received July 28, 2004     

Characterizing path graphs by forbidden induced subgraphs

A path graph is the intersection graph of subpaths of a tree. In 1970, Renz asked for a characterization of path graphs by forbidden induced subgraphs. We answer this question by determining the complete list of graphs that are not path graphs and are minimal with this property. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 369–384, 2009     

Fast Enumeration Algorithms for Non-crossing Geometric Graphs

A non-crossing geometric graph is a graph embedded on a set of points in the plane with non-crossing straight line segments. In this paper we present a general framework for enumerating non-crossing geometric graphs on a given point set. Applying our idea to specific enumeration problems, we obtain faster algorithms for enumerating plane straight-line graphs, non-crossing spanning connected graphs, non-crossing spanning trees, and non-crossing minimally rigid graphs. Our idea also produces efficient enumeration algorithms for other graph classes, for which no algorithm has been reported so far, such as non-crossing matchings, non-crossing red-and-blue matchings, non-crossing k-vertex or k-edge connected graphs, or non-crossing directed spanning trees. The proposed idea is relatively simple and potentially applies to various other problems of non-crossing geometric graphs.     

On Spin Models,Triply Regular Association Schemes,and Duality

Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using series-parallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.     

Digraph matrix partitions and trigraph homomorphisms

Matrix partitions generalize graph colourings and homomorphisms. Their study has so far been confined to symmetric matrices and undirected graphs. In this paper we make an initial study of list matrix partitions for digraphs; in other words our matrices are not necessarily symmetric. We motivate future conjectures by classifying the complexity of all list matrix partition problems for matrices of size up to three. We find it convenient to model the problem in the language of trigraph homomorphisms.     

Some characterizations of doubly chordal graphs

Many optimization problems like domination and Steiner tree are NP-complete onchordal graphs but can be solved in polynomial time ondoubly chordal graphs. Investigating properties of doubly chordal graphs probably help to design efficient algorithms for the graphs. We present some characterizations of doubly chordal graphs, which are based on clique matrices and neighborhood matrices. It is also mentioned how adoubly perfect elimination ordering of a doubly chordal graph can be computed from the results.     

Line graphs: Their maximum nullities and zero forcing numbers

The maximum nullity over a collection of matrices associated with a graph has been attracting the attention of numerous researchers for at least three decades. Along these lines various zero forcing parameters have been devised and utilized for bounding the maximum nullity. The maximum nullity and zero forcing number, and their positive counterparts, for general families of line graphs associated with graphs possessing a variety of specific properties are analysed. Building upon earlier work, where connections to the minimum rank of line graphs were established, we verify analogous equations in the positive semidefinite cases and coincidences with the corresponding zero forcing numbers. Working beyond the case of trees, we study the zero forcing number of line graphs associated with certain families of unicyclic graphs.