Mixed Graph Research Articles

Analogues of Cliques for (m, n)-Colored Mixed Graphs

An (m,n)-colored mixed graph is a mixed graph with arcs assigned one of m different colors and edges one of n different colors. A homomorphism of an (m,n)-colored mixed graph G to an (m,n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is also an arc (edge) of color c. The (m,n)-colored mixed chromatic number, denoted chi_{m,n}(G), of an (m,n)-colored mixed graph G is the order of a smallest homomorphic image of G. An (m,n)-clique is an (m,n)-colored mixed graph C with chi_{m,n}(C) = |V(C)|. Here we study the structure of (m,n)-cliques. We show that almost all (m,n)-colored mixed graphs are (m,n)-cliques, prove bounds for the order of a largest outerplanar and planar (m,n)-clique and resolve an open question concerning the computational complexity of a decision problem related to (0,2)-cliques. Additionally, we explore the relationship between chi_{1,0} and chi_{0,2}.

The Hierarchical Mixed Rural Postman Problem

In this paper, we study a generalization of the Hierarchical Chinese Postman Problem on a mixed graph where only a subset of arcs and edges require a service to be accomplished following a hierarchical order. The problem, called the Hierarchical Mixed Rural Postman Problem, also generalizes the Rural Postman Problem and thus is NP-hard. We propose a new mathematical formulation, and introduce two effective solution algorithms. The first procedure is a matheuristic that is based on the exact solution of a variant of the Mixed Rural Postman Problem for each hierarchy. The second approach is a tabu search algorithm based on different improvement and diversification strategies. Computational results on an extended set of instances show how the proposed solution methods are quite effective and efficient when compared to the solutions of a branch-and-cut algorithm stopped after one hour of computation.

Principal minor version of Matrix-Tree theorem for mixed graphs

In Yu, et al. (2017), an analytical expression of the determinant of the Hermitian (quasi-)Laplacian matrix of mixed graphs has been proven. In this paper, we are going to extend those results and derive an analytical expression for the principal minors of the Hermitian (quasi-)Laplacian matrix, which is the principal minor version of the Matrix-Tree theorem.

Mixed graphs with H-rank 3

In the article (Mohar, 2016 [9]), Mohar determined all mixed graphs with H-rank 2, and used it to classify cospectral graphs with respect to their Hermitian adjacency matrix, constructing a class of graphs which can not be determined by their H-spectrum. In the present paper, we investigate the H-rank of mixed graphs further, determining the H-ranks of those mixed graphs with trees, cycles and complete bipartite graphs as underlying graphs, respectively. Moreover, we characterize all mixed graphs with H-rank 3, and show that all connected mixed graphs with H-rank 3 can be determined by their H-spectrum.

Hermitian-Randi\u0107 matrix and Hermitian-Randi\u0107 energy of mixed graphs

Let M be a mixed graph and H(M) be its Hermitian-adjacency matrix. If we add a Randić weight to every edge and arc in M, then we can get a new weighted Hermitian-adjacency matrix. What are the properties of this new matrix? Motivated by this, we define the Hermitian-Randić matrix R_{H}(M)=(r_{h})_{kl} of a mixed graph M, where (r_{h})_{kl}=-(r_{h})_{lk}=frac{mathbf{i}}{sqrt {d_{k}d_{l}}} (mathbf{i}=sqrt{-1}) if (v_{k},v_{l}) is an arc of M, (r_{h})_{kl}=(r_{h})_{lk}=frac{1}{sqrt{d_{k}d_{l}}} if v_{k}v_{l} is an undirected edge of M, and (r_{h})_{kl}=0 otherwise. In this paper, firstly, we compute the characteristic polynomial of the Hermitian-Randić matrix of a mixed graph. Furthermore, we give bounds on the Hermitian-Randić energy of a general mixed graph. Finally, we give some results about the Hermitian-Randić energy of mixed trees.

PAG2ADMG: A Novel Methodology to Enumerate Causal Graph Structures

Causal graphs, such as directed acyclic graphs (DAGs) and partial ancestral graphs (PAGs), represent causal relationships among variables in a model. Methods exist for learning DAGs and PAGs from data and for converting DAGs to PAGs. However, these methods only output a single causal graph consistent with the independencies/dependencies (the Markov equivalence class M) estimated from the data. However, many distinct graphs may be consistent with M, and a data modeler may wish to select among these using domain knowledge. In this paper, we present a method that makes this possible. We introduce PAG2ADMG, the first method for enumerating all causal graphs consistent with M, under certain assumptions. PAG2ADMG converts a given PAG into a set of acyclic directed mixed graphs (ADMGs). We prove the correctness of the approach and demonstrate its efficiency relative to brute-force enumeration.

The Enhanced Principal Rank Characteristic Sequence for Hermitian Matrices

The enhanced principal rank characteristic sequence (epr-sequence) of an $n\x n$ matrix is a sequence $\ell_1 \ell_2 \cdots \ell_n$, where each $\ell_k$ is ${\tt A}$, ${\tt S}$, or ${\tt N}$ according as all, some, or none of its principal minors of order $k$ are nonzero. There has been substantial work on epr-sequences of symmetric matrices (especially real symmetric matrices) and real skew-symmetric matrices, and incidental remarks have been made about results extending (or not extending) to (complex) Hermitian matrices. A systematic study of epr-sequences of Hermitian matrices is undertaken; the differences with the case of symmetric matrices are quite striking. Various results are established regarding the attainability by Hermitian matrices of epr-sequences that contain two ${\tt N}$s with a gap in between. Hermitian adjacency matrices of mixed graphs that begin with ${\tt NAN}$ are characterized. All attainable epr-sequences of Hermitian matrices of orders $2$, $3$, $4$, and $5$, are listed with justifications.

The spectral distribution of random mixed graphs

Let G be a mixed graph with n vertices, H(G) the Hermitian adjacency matrix of G, and λ1(G),λ2(G),…,λn(G) the eigenvalues of H(G). The Hermitian energy of G is defined as EH(G)=∑i=1n|λi(G)|. In this paper we characterize the limiting spectral distribution of the Hermitian adjacency matrices of random mixed graphs, and as an application, we give an estimation of the Hermitian energy for almost all mixed graphs.

Empirical likelihood for linear structural equation models with dependent errors

We consider linear structural equation models that are associated with mixed graphs. The structural equations in these models only involve observed variables, but their idiosyncratic error terms are allowed to be correlated and non‐Gaussian. We propose empirical likelihood procedures for inference and suggest several modifications, including a profile likelihood, in order to improve tractability and performance of the resulting methods. Through simulations, we show that when the error distributions are non‐Gaussian, the use of empirical likelihood and the proposed modifications may increase statistical efficiency and improve assessment of significance. Copyright © 2017 John Wiley & Sons, Ltd.

Sequence mixed graphs

A mixed graph can be seen as a type of digraph containing some edges (or two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and iterated line digraphs. These structures are proven to be useful in the problem of constructing dense graphs or digraphs, and this is related to the degree/diameter problem. Thus, our generalized approach gives rise to graphs that have also good ratio order/diameter. Moreover, we propose a general method for obtaining a sequence mixed digraph by identifying some vertices of a certain iterated line digraph. As a consequence, some results about distance-related parameters (mainly, the diameter and the average distance) of sequence mixed graphs are presented.

A family of mixed graphs with large order and diameter 2

A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order.We present a construction that provides mixed graphs of undirected degree q, directed degree q−12 and order 2q2, for q being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to 9q2−4q+34 the defect of these mixed graphs is (q−22)2−14.In particular we obtain a known mixed Moore graph of order 18, undirected degree 3 and directed degree 1 called Bosák’s graph and a new mixed graph of order 50, undirected degree 5 and directed degree 2, which is proved to be optimal.

Constraint graphs for combinatorial mobility determination

The mobility is a fundamental property of mechanisms and any design methodology needs to estimate at least the generic topological mobility, i.e. the mobility that almost all mechanisms with a given kinematic topology exhibit. The determination of the topological mobility requires combinatorial algorithms operating upon a graph representation of the kinematic constraints. A faithful graph representation is thus crucial. The lack of such a graph representation was a key obstacle for applying combinatorial methods.As solution to problem, in this paper a general kinematic constraint graph, called elementary graph (EG), is introduced allowing for a unique representation of constraints including multiple-joints. From this a reduced graph is derived, called mixed graph (MG), which contains the minimal number of elements while ensuring uniqueness. It shown that from the MG can be derived the well-known body-bar (BB) and bar-joint (BJ) graphs as special cases.The novel MG provides a unique basis for combinatorial algorithms for mobility determination. The most efficient algorithm is the Pebble Game, widely used in biology and chemistry, which exists for BB and BJ. Using the MG requires an adapted variant of the Pebble Game. This will be reported in a forthcoming paper.As prerequisite, the terminology used to describe the mobility of mechanisms is reviewed and a consistent general concept for topological mobility is introduced. Also the shortcomings of structure mobility criteria are briefly discussed.

A construction of dense mixed graphs of diameter 2

A mixed graph is said to be dense, if its order is close to the Moore bound and it is optimal if there is not a mixed graph with the same parameters and bigger order. We give a construction that provides dense mixed graphs of undirected degree q, directed degree q−12 and order 2q2, for q being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to 9q2−4q+34 the defect of these mixed graphs is (q−22)2−14. In particular we obtain a known mixed Moore graph of order 18, undirected degree 3 and directed degree 1, called Bosák's graph and a new mixed graph of order 50, undirected degree 5 and directed degree 2, which is proved to be optimal.

A variant of the McKay-Miller-Širáň construction for mixed graphs

The Degree/Diameter Problem is an extremal problem in graph theory with applications in network design. One of the main research areas in the Degree/Diameter Problem consists of finding large graphs whose order approach the theoretical upper bounds as much as possible. In the case of directed graphs there exist some families that come close to the theoretical upper bound asymptotically. In the case of undirected graphs there also exist some good constructions for specific values of the parameters involved (degree and/or diameter). One such construction was given by McKay, Miller, and Širáň in [McKay, B., M. Miller and J. Širáň, A note on large graphs of diameter two and given maximum degree, J Comb Theo Ser B 74 (1998), 110-118], which produces large graphs of diameter 2 with the aid of the voltage assignment technique. Here we show how to re-engineer the McKay-Miller-Širáň construction in order to obtain large mixed graphs of diameter 2, i.e. graphs containing both directed arcs and undirected edges.

Chinese Postman Problem on edge-colored multigraphs

It is well-known that the Chinese Postman Problem on undirected and directed graphs is polynomial-time solvable. We extend this result to edge-colored multigraphs. Our result is in sharp contrast to the Chinese Postman Problem on mixed graphs, i.e., graphs with directed and undirected edges, for which the problem is NP-hard.

Parameterized complexity of the k-arc Chinese Postman Problem

In the Mixed Chinese Postman Problem (MCPP), given an edge-weighted mixed graph G (G may have both edges and arcs), our aim is to find a minimum weight closed walk traversing each edge and arc at least once. The MCPP parameterized by the number of edges was known to be fixed-parameter tractable using a simple argument. Solving an open question of van Bevern et al., we prove that the MCPP parameterized by the number of arcs is also fixed-parameter tractable. Our proof is more involved and, in particular, uses a well-known result of Marx, O'Sullivan and Razgon (2013) on the treewidth of torso graphs with respect to small separators. We obtain a small cut analog of this result, and use it to construct a tree decomposition which, despite not having bounded width, has other properties allowing us to design a fixed-parameter algorithm.

Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs

A mixed graph is obtained from an undirected graph by orienting a subset of its edges. The Hermitian adjacency matrix of a mixed graph M of order n is an n × n matrix H(M)=(hkl), where hkl=−hlk=i (i=−1) if there exists an orientation from vk to vl and hkl=hlk=1 if there exists an edge between vk and vl but not exist any orientation, and hkl=0 otherwise. Let D(M)=diag(d1,d2,…,dn) be a diagonal matrix where di is the degree of vertex vi in the underlying graph Mu. Hermitian matrices L(M)=D(M)−H(M),Q(M)=D(M)+H(M) are said as the Hermitian Laplacian matrix, Hermitian quasi-Laplacian matrix of mixed graph M, respectively. In this paper, it is shown that they are positive semi-definite. Moreover, we characterize the singularity of them. In addition, an expression of the determinant of the Hermitian (quasi-)Laplacian matrix is obtained.

First Eigenvalue of Nonsingular Mixed Unicyclic Graphs with Fixed Number of Branch Vertices

Mixed graphs are graphs whose edges may be directed or undirected. The first eigenvalue of a mixed graph is the least nonzero eigenvalue of its Laplacian matrix. We determine the unique mixed graphs with minimum first eigenvalue over all nonsingular mixed unicyclic graphs with fixed number of branch vertices, and the unique graph with minimum least signless Laplacian eigenvalue over all nonbipartite unicyclic graphs with fixed number of branch vertices.

Mixed graph colouring for unit-time scheduling

We consider the job shop scheduling problem with unit-time operations and the makespan criterion. This problem is reduced to finding an optimal colouring of a special class of mixed graph, where its partial graph without edges represents the union of maximal directed paths and its partial graph without arcs represents the union of maximal cliques. As the problem is known to be NP-hard, both exact and heuristic methods are proposed to solve it. This study is carried out in three steps. First, a new lower and upper bounds for the mixed chromatic number are proposed. Afterwards, a colour-indexed mathematical model using the proposed bounds is developed. Then, a tabu search using a dynamic neighbourhood structure is adapted for solving large instances. Computational experiments conducted on several modified benchmarks show the efficiency and effectiveness of the proposed resolution methods.

Marginalization and conditioning for LWF chain graphs

In this paper, we deal with the problem of marginalization over and conditioning on two disjoint subsets of the node set of chain graphs (CGs) with the LWF Markov property. For this purpose, we define the class of chain mixed graphs (CMGs) with three types of edges and, for this class, provide a separation criterion under which the class of CMGs is stable under marginalization and conditioning and contains the class of LWF CGs as its subclass. We provide a method for generating such graphs after marginalization and conditioning for a given CMG or a given LWF CG. We then define and study the class of anterial graphs, which is also stable under marginalization and conditioning and contains LWF CGs, but has a simpler structure than CMGs.