Mixed Graph Research Articles

On the characteristic polynomial and energy of Hermitian quasi-Laplacian matrix of mixed graphs

A mixed graph is a graph whose edge set consists of both oriented and unoriented edges. The Hermitian-adjacency matrix of an [Formula: see text]-vertex mixed graph is a square matrix [Formula: see text] of order [Formula: see text], where [Formula: see text] if there is an arc from [Formula: see text] to [Formula: see text] and [Formula: see text] if there is an edge between [Formula: see text] and [Formula: see text], and [Formula: see text] otherwise. Let [Formula: see text] be a diagonal matrix, where [Formula: see text] is the degree of [Formula: see text] in the underlying graph of [Formula: see text]. The matrices [Formula: see text] and [Formula: see text] are, respectively, the Hermitian Laplacian and Hermitian quasi-Laplacian matrix of the mixed graph [Formula: see text]. In this paper, we first found coefficients of the characteristic polynomial of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph [Formula: see text]. Second, we discussed relationship between the spectra of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph [Formula: see text].

Nested Markov properties for acyclic directed mixed graphs

Conditional independence models associated with directed acyclic graphs (DAGs) may be characterized in at least three different ways: via a factorization, the global Markov property (given by the d-separation criterion), and the local Markov property. Marginals of DAG models also imply equality constraints that are not conditional independences; the well-known “Verma constraint” is an example. Constraints of this type are used for testing edges, and in a computationally efficient marginalization scheme via variable elimination. We show that equality constraints like the “Verma constraint” can be viewed as conditional independences in kernel objects obtained from joint distributions via a fixing operation that generalizes conditioning and marginalization. We use these constraints to define, via ordered local and global Markov properties, and a factorization, a graphical model associated with acyclic directed mixed graphs (ADMGs). We prove that marginal distributions of DAG models lie in this model, and that a set of these constraints given by Tian provides an alternative definition of the model. Finally, we show that the fixing operation used to define the model leads to a particularly simple characterization of identifiable causal effects in hidden variable causal DAG models.

Incidence matrices and line graphs of mixed graphs

Abstract In the theory of line graphs of undirected graphs, there exists an important theorem linking the incidence matrix of the root graph to the adjacency matrix of its line graph. For directed or mixed graphs, however, there exists no analogous result. The goal of this article is to present aligned definitions of the adjacency matrix, the incidence matrix, and line graph of a mixed graph such that the mentioned theorem is valid for mixed graphs.

HS-integral and Eisenstein integral mixed circulant graphs

A mixed graph is called \emph{second kind hermitian integral} (\emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of the second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set $S$ for which a mixed circulant graph $\text{Circ}(\mathbb{Z}_n, S)$ is HS-integral. We also show that a mixed circulant graph is Eisenstein integral if and only if it is HS-integral. Further, we express the eigenvalues and the HS-eigenvalues of unitary oriented circulant graphs in terms of generalized M$\ddot{\text{o}}$bius function.

The gamma-Signless Laplacian Adjacency Matrix of Mixed Graphs

The gamma-Signless Laplacian Adjacency Matrix of Mixed Graphs

Cycle-connected mixed graphs and related problems

Cycle-connected mixed graphs and related problems

On clique numbers of colored mixed graphs

An (m,n)-colored mixed graph, or simply, an (m,n)-graph is a graph having m different types of arcs and n different types of edges. A homomorphism of an (m,n)-graph G to another (m,n)-graph H is a vertex mapping that preserves adjacency, the type thereto and the direction. A subset R of the set of vertices of G that always maps distinct vertices in itself to distinct image vertices under any homomorphism is called an (m,n)-relative clique of G. The maximum cardinality of an (m,n)-relative clique of a graph is called the (m,n)-relative clique number of the graph. In this article, we explore the (m,n)-relative clique numbers for various families of graphs.

Resonance algorithm: an intuitive algorithm to find all shortest paths between two nodes

The shortest path problem (SPP) is a classic problem and appears in a wide range of applications. Although a variety of algorithms already exist, new advances are still being made, mainly tuned for particular scenarios to have better performances. As a result, they become more and more technically complex and sophisticated. In this paper, we developed an intuitive and nature-inspired algorithm to compute all possible shortest paths between two nodes in a graph: Resonance Algorithm (RA). It can handle any undirected, directed, or mixed graphs, irrespective of loops, unweighted or positively weighted edges, and can be implemented in a fully decentralized manner. Although the original motivation for RA is not the speed per se, in certain scenarios (when sophisticated matrix operations can be employed, and when the map is very large and all possible shortest paths are demanded), it out-competes Dijkstra’s algorithm, which suggests that in those scenarios, RA could also be practically useful.

IT ARCHITECTURE SOLUTIONS FOR INDUSTRY BASED COOPERATIONS INTEGRATION SUPPORT

The current research provides solution for the process of legal cooperation between enterprises of participants according to the technological and delivery principle information support. As a solution, it is proposed to design a single information space based on existing information systems using microservice architecture technologies. This article discusses the main problems of supporting the integration process of intra-industry cooperation, provides a model for the integral assessment of the industry cooperation effectiveness based on the method of Saaty hierarchies and expert assessments, the article describes the contract of interaction within the framework of the integration process. The article also presents an architecture model built using the object-oriented language UML, provides an assessment of the system stability of the architecture model using a mixed graph, and presents the results of an architecture’s practical assessment.

The no-meet matroid

Given a digraph and a subset of vertices representing initial positions, we study the existence problem of infinitely long walks, one starting from each initial position, that never meet. We show that the subsets for which this is possible constitute the independent sets of a matroid. We prove that the independence oracle is polynomial-time. We also provide a more efficient algorithm to compute the size of a basis. Then we focus on the orientation problem of the undirected edges of a mixed graph to either maximize or minimize the rank of a subset. Some NP-hardness results and inapproximability results are proved in the general case. Polynomial-time algorithms are described for subsets of size 1. We also report several connections with other matroid classes. For example, we show that the class of no-meet matroids strictly contains transversal matroids while it is strictly contained inside gammoids. Some extensions and related applications are also discussed.

Principal Minors of Hermitian (Quasi-)Laplacian Matrix of Second Kind for Mixed Graphs

Principal Minors of Hermitian (Quasi-)Laplacian Matrix of Second Kind for Mixed Graphs

Computing maximum likelihood estimates for Gaussian graphical models with Macaulay2

We introduce the package "GraphicalModelsMLE" for computing the maximum likelihood estimates (MLEs) of a Gaussian graphical model in the computer algebra system Macaulay2. This package allows the computation of MLEs for the class of loopless mixed graphs. Additional functionality allows the user to explore the underlying algebraic structure of the model, such as its maximum likelihood degree and the ideal of score equations.

The k-generalized Hermitian adjacency matrices for mixed graphs

This article gives some fundamental introduction to spectra of mixed graphs via its k-generalized Hermitian adjacency matrix. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the kth root of unity e2πik (and its symmetric entry is e−2πik); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. For all positive integers k, the non-zero entries of the above matrix are chosen from the gain set {1,e2πik,e−2πik}, which is not closed under multiplication when k⩾4. In this paper, for all positive integers k, we extract all the mixed graphs whose k-generalized Hermitian adjacency rank (Hk-rank for short) is 3, which partially answers a question proposed by Wissing and van Dam [34]. Furthermore, we study the spectral determination of mixed graphs with Hk-rank 2 and 3, respectively.

The 2-colouring problem for $(m,n)$-mixed graphs with switching is polynomial

A mixed graph is a set of vertices together with an edge set and an arc set. An $(m,n)$-mixed graph $G$ is a mixed graph whose edges are each assigned one of $m$ colours, and whose arcs are each assigned one of $n$ colours. A \emph{switch} at a vertex $v$ of $G$ permutes the edge colours, the arc colours, and the arc directions of edges and arcs incident with $v$. The group of all allowed switches is $\Gamma$. Let $k \geq 1$ be a fixed integer and $\Gamma$ a fixed permutation group. We consider the problem that takes as input an $(m,n)$-mixed graph $G$ and asks if there a sequence of switches at vertices of $G$ with respect to $\Gamma$ so that the resulting $(m,n)$-mixed graph admits a homomorphism to an $(m,n)$-mixed graph on $k$ vertices. Our main result establishes this problem can be solved in polynomial time for $k \leq 2$, and is NP-hard for $k \geq 3$. This provides a step towards a general dichotomy theorem for the $\Gamma$-switchable homomorphism decision problem.

Fair integral submodular flows

Integer-valued elements of an integral submodular flow polyhedron Q are investigated which are decreasingly minimal (dec-min) in the sense that their largest component is as small as possible, within this, the second largest component is as small as possible, and so on. As a main result, we prove that the set of dec-min integral elements of Q is the set of integral elements of another integral submodular flow polyhedron arising from Q by intersecting a face of Q with a box. Based on this description, we develop a strongly polynomial algorithm for computing not only a dec-min integer-valued submodular flow but even a cheapest one with respect to a linear cost-function. A special case is the problem of finding a strongly connected (or k -edge-connected) orientation of a mixed graph whose in-degree vector is decreasingly minimal.

A two-phase hybrid algorithm for the periodic rural postman problem with irregular services on mixed graphs

In this article we address the periodic rural postman problem with irregular services (PRPP–IS), where some arcs and/or edges of a mixed graph must be traversed (to be serviced) a certain number of times in some subsets of days of a given time horizon. The goal is to define a set of minimum cost tours, one for each day or period of the time horizon, that satisfy the service requirements. For this problem we propose a two-phase algorithm that combines heuristics and mathematical programming. In the first phase, two different procedures are used to construct feasible solutions: a multi-start heuristic based on feasibility pump and a multi-start constructive heuristic. From these solutions, some fragments (parts of tours associated with the different days) are extracted. The second phase determines a solution for the PRPP–IS by combining the fragments by means of a mathematical model. We show the effectiveness of this solution approach through an extensive experimental phase on different sets of instances.

On weight-symmetric 3-coloured digraphs

We consider particular weighted directed graphs with edges having colour red, blue or green such that each red edge has weight 1, each blue edge has weight −1 and each green edge has weight i (the imaginary unit). Such a directed graph is called a 3-coloured digraph (for short, a 3-CD). Every mixed graph and every signed graph can be interpreted as a 3-CD. We first study some structural properties of 3-CDs by means of the eigenvalues of the adjacency matrix. In particular, we give spectral criteria for singularity of such digraphs. Second, we consider weight-symmetric 3-CDs, i.e. those 3-CDs that are switching isomorphic to their negation. It follows that the class of weight-symmetric 3-CDs is included in the class of 3-CDs whose spectrum is symmetric (with respect to the origin). We give some basic properties and several constructions of weight-symmetric 3-CDs and establish constructions of 3-CDs which have symmetric spectrum but are not weight-symmetric.

On -gain graphs with few positive eigenvalues

Let T4={1,−1,i,−i} be the group of fourth roots of unit. A T4-gain graph is a graph where each orientation of an edge is given a complex unit in T4, which is the inverse of the complex unit assigned to the opposite orientation. In this paper, we characterize the structure of the T4-gain graphs with exactly one positive eigenvalue and determine the T4-gain graphs with cut vertices having exactly two positive eigenvalues. Our results extend some parallel ones about mixed graphs and signed graphs.

Integral mixed circulant graphs

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integers. The mixed circulant graphCirc(Zn,C) is a mixed graph on the vertex set Zn and edge set {(a,b):b−a∈C}, where 0∉C⊂Zn. If C is closed under inverse, then Circ(Zn,C) is called a circulant graph. We express the eigenvalues of Circ(Zn,C) in terms of primitive n-th roots of unity, and find a sufficient condition for integrality of the eigenvalues of Circ(Zn,C). For n≡0(mod4), we factorize the n-th degree cyclotomic polynomial into two irreducible factors over Q(i). Using this factorization, we characterize integral mixed circulant graphs in terms of their symbol set. We also express the integer eigenvalues of an integral oriented circulant graph in terms of a Ramanujan type sum, and discuss some of their properties.

The spectra of random mixed graphs

A mixed graph is a graph that can be obtained from a simple undirected graph by replacing some of the edges by arcs in precisely one of the two possible directions. The Hermitian adjacency matrix of a mixed graph G of order n is the n×n matrix H(G)=(hij), where hij=−hji=i (with i=−1) if there exists an arc from vi to vj (but no arc from vj to vi), hij=hji=1 if there exists an edge (and no arcs) between vi and vj, and hij=hji=0 otherwise (if vi and vj are neither joined by an edge nor by an arc). Let λ1(G),λ2(G),…,λn(G) be eigenvalues of H(G). The k-th Hermitian spectral moment of G is defined as sk(H(G))=∑i=1nλik(G), where k≥0 is an integer. In this paper, we deal with the asymptotic behavior of the spectrum of the Hermitian adjacency matrix of random mixed graphs. We will present and prove a separation result between the largest and remaining eigenvalues of the Hermitian adjacency matrix, and as an application, we estimate the Hermitian spectral moments of random mixed graphs.