Mixed Graph Research Articles

Bivariate Chromatic Polynomials of Mixed Graphs

The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses $\chi_G(x,y)$ as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for $\chi_G(x,y)$.

On spectral invariants of the [formula omitted]-mixed adjacency matrix

Let Gˆ be a mixed graph and α∈[0,1]. Let Dˆ(Gˆ) and Aˆ(Gˆ) be the diagonal matrix of vertex degrees and the mixed adjacency matrix of Gˆ, respectively. The α-mixed adjacency matrix of Gˆ is the matrix Aˆα(Gˆ)=αDˆ(Gˆ)+(1−α)Aˆ(Gˆ).We study some properties of Aˆα(Gˆ) associated with some type of mixed graphs, namely quasi-bipartite and pre-bipartite mixed graphs. A spectral characterization for pre-bipartite and some class of quasi-bipartite mixed graphs is given. For a mixed graph Gˆ we exploit the problem of finding the smallest α for which Aˆα(Gˆ) is positive semi-definite. This problem was proposed by Nikiforov in the context of undirected graphs. It is proven here that, for a mixed graph this number is not greater than 12 and that a connected mixed graph Gˆ with n≥2 is quasi-bipartite if and only if this number is exactly 12. The spread of the α-mixed adjacency matrix is the difference among the largest and the smallest α-mixed adjacency eigenvalue. Upper and lower bounds for the spread of the α- mixed adjacency matrix are obtained. The α-mixed Estrada index of Gˆ is the sum of the exponentials of the eigenvalues of Aˆα(Gˆ). In this paper, bounds for the eigenvalues of Aˆα(Gˆ) are established and, using these bounds some sharp bounds on the mixed Estrada index of Aˆα(Gˆ) are presented.

Graphical tools for selecting conditional instrumental sets

Summary We consider the efficient estimation of total causal effects in the presence of unmeasured confounding using conditional instrumental sets. Specifically, we consider the two-stage least-squares estimator in the setting of a linear structural equation model with correlated errors that is compatible with a known acyclic directed mixed graph. To set the stage for our results, we characterize the class of linearly valid conditional instrumental sets that yield consistent two-stage least-squares estimators for the target total effect and derive a new asymptotic variance formula for these estimators. Equipped with these results, we provide three graphical tools for selecting more efficient linearly valid conditional instrumental sets: first, a graphical criterion that, for certain pairs of linearly valid conditional instrumental sets, identifies which of the two corresponding estimators has the smaller asymptotic variance second, an algorithm that greedily adds covariates that reduce the asymptotic variance to a given linearly valid conditional instrumental set and, third, a linearly valid conditional instrumental set for which the corresponding estimator has the smallest asymptotic variance that can be ensured with a graphical criterion.

Packing mixed hyperarborescences

The aim of this paper is twofold. We first provide a new orientation theorem which gives a natural and simple proof of a result of Gao and Yang (2021) on matroid-reachability-based packing of mixed arborescences in mixed graphs by reducing it to the corresponding theorem of Király (2016) on directed graphs. Moreover, we extend another result of Gao and Yang (2021) by providing a new theorem on mixed hypergraphs having a packing of mixed hyperarborescences such that their number is at least ℓ and at most ℓ′, each vertex belongs to exactly k of them, and each vertex v is the root of least f(v) and at most g(v) of them.

Highly robust spatiotemporal wavefront prediction with a mixed graph neural network in adaptive optics

The time-delay problem, which is introduced by the response time of hardware for correction, is a critical and non-ignorable problem of adaptive optics (AO) systems. It will result in significant wavefront correction errors while turbulence changes severely or system responses slowly. Predictive AO is proposed to alleviate the time-delay problem for more accurate and stable corrections in the real time-varying atmosphere. However, the existing prediction approaches either lack the ability to extract non-linear temporal features, or overlook the authenticity of spatial features during prediction, leading to poor robustness in generalization. Here, we propose a mixed graph neural network (MGNN) for spatiotemporal wavefront prediction. The MGNN introduces the Zernike polynomial and takes its inherent covariance matrix as physical constraints. It takes advantage of conventional convolutional layers and graph convolutional layers for temporal feature catch and spatial feature analysis, respectively. In particular, the graph constraints from the covariance matrix and the weight learning of the transformation matrix promote the establishment of a realistic internal spatial pattern from limited data. Furthermore, its prediction accuracy and robustness to varying unknown turbulences, including the generalization from simulation to experiment, are all discussed and verified. In experimental verification, the MGNN trained with simulated data can achieve an approximate effect of that trained with real turbulence. By comparing it with two conventional methods, the demonstrated performance of the proposed method is superior to the conventional AO in terms of root mean square error (RMS). With the prediction of the MGNN, the mean and standard deviation of RMS in the conventional AO are reduced by 54.2% and 58.6% at most, respectively. The stable prediction performance makes it suitable for wavefront predictive correction in astronomical observation, laser communication, and microscopic imaging.

A forecasting framework on fusion of spatiotemporal features for multi-station PM2.5

Persistent PM2.5 pollution poses a serious threat to human health. Developing an accurate urban regional PM2.5 forecasting is of practical significance for environmental protection. However, previous studies have mostly focused on individual monitoring stations, neglecting the influence of neighboring stations, which limits forecasting accuracy. Additionally, the PM2.5 of a single monitoring station cannot reflect the overall situation of a region. Therefore, this paper develops a novel PM2.5 spatiotemporal forecasting framework that combines graph convolutional module, temporal convolutional module, linear module. It enables the forecasting of PM2.5 concentrations at multiple stations and multiple time steps in the future. Concretely, we utilize a mixed graph convolutional network to extract the spatial features of PM2.5. Then, an improved temporal convolutional network, the second-order residual temporal convolutional network, is developed to capture complex temporal features. Following the classical “linear and non-linear” modeling strategy, a linear module is added to the forecasting framework. Experiments on the real air pollution dataset from Beijing demonstrate that our framework outperforms the state-of-the-art baselines.

H-integral and Gaussian integral normal mixed Cayley graphs

If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called H-integral. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed graph are Gaussian integers, then the mixed graph is called Gaussian integral. For any finite group Γ, we characterize the set S for which the normal mixed Cayley graph Cay(Γ,S) is H-integral. We further prove that a normal mixed Cayley graph is H-integral if and only if the mixed graph is Gaussian integral.

Mixed graphs whose Hermitian adjacency matrices of the second kind have the smallest eigenvalue greater than [formula omitted

Mixed graphs unify simple graphs and oriented graphs naturally. The Hermitian adjacency matrix of a mixed graph for the second kind (N-matrix for short) was recently introduced by Mohar [13]. 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 principal sixth root of unity ω=1+i32 (and its symmetric entry is ω‾=1−i32); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. In this paper, we completely characterize the connected mixed graphs whose N-matrices have the smallest eigenvalue greater than −32. It may be considered as the continuance of the work of J.H. Smith [18] who considered the classical problem of characterizing the graphs whose eigenvalues lie in a given interval.

Complexity of (arc)-connectivity problems involving arc-reversals or deorientations

By a well known theorem of Robbins, a graph G has a strongly connected orientation if and only if G is 2-edge-connected and it is easy to find, in linear time, either a cut edge of G or a strong orientation of G. A result of Durand de Gevigney shows that for every k≥3 it is NP-hard to decide if a given graph G has a k-strong orientation. Thomassen showed that one can check in polynomial time whether a given graph has a 2-strong orientation. This implies that for a given digraph D we can determine in polynomial time whether we can reorient (=reverse) some arcs of D=(V,A) to obtain a 2-strong digraph D′=(V,A′). This naturally leads to the question of determining the minimum number of such arcs to reverse before the resulting graph is 2-strong. In this paper we show that finding this number is NP-hard. If a 2-connected graph G has no 2-strong orientation, we may ask how many of its edges we may orient so that the resulting mixed graph is still 2-strong. Similarly, we may ask for a 2-edge-connected graph G how many of its edges we can orient such that the resulting mixed graph remains 2-arc-strong. We prove that when restricted to graphs satisfying suitable connectivity conditions, both of these problems are equivalent to finding the minimum number of edges we must double in a 2-edge-connected graph in order to obtain a 4-edge-connected graph. Using this, we show that all these three problems are NP-hard.Finally, we consider the operation of deorienting an arc uv of a digraph D meaning replacing it by an undirected edge between the same vertices. In terms of connectivity properties, this is equivalent to adding the opposite arc vu to D. We prove that for every ℓ≥3 it is NP-hard to find the minimum number of arcs to deorient in a digraph D in order to obtain an ℓ-strong digraph D′.

HS-splitting fields of abelian mixed cayley graphs

For an n-vertex mixed graph A, let HS(A) be the Hermitian-adjacency matrix of the second kind of A and ΦA(HS,λ)=det(λIn−HS(A)) the characteristic polynomial of HS(A). The splitting field of ΦA(HS,λ) is referred to as the HS-splitting field of A. Its extension degree over the rational number field Q is referred to as the HS-algebraic degree of A, and A is said to be HS-integral if all eigenvalues of HS(A) are integers. In this paper, we give explicit expressions for the HS-splitting fields of abelian mixed Cayley graphs. In addition, we derive a formula to calculate their corresponding HS-algebraic degrees. Moreover, we characterize all HS-integral abelian mixed Cayley graphs.

The establishment and validation of a prediction model for traumatic intracranial injury patients: a reliable nomogram.

Traumatic brain injury (TBI) leads to death and disability. This study developed an effective prognostic nomogram for assessing the risk factors for TBI mortality. Data were extracted from an online database called "Multiparameter Intelligent Monitoring in Intensive Care IV" (MIMIC IV). The ICD code obtained data from 2,551 TBI persons (first ICU stay, >18 years old) from this database. R divided samples into 7:3 training and testing cohorts. The univariate analysis determined whether the two cohorts differed statistically in baseline data. This research used forward stepwise logistic regression after independent prognostic factors for these TBI patients. The optimal variables were selected for the model by the optimal subset method. The optimal feature subsets in pattern recognition improved the model prediction, and the minimum BIC forest of the high-dimensional mixed graph model achieved a better prediction effect. A nomogram-labeled TBI-IHM model containing these risk factors was made by nomology in State software. Least Squares OLS was used to build linear models, and then the Receiver Operating Characteristic (ROC) curve was plotted. The TBI-IHM nomogram model's validity was determined by receiver operating characteristic curves (AUCs), correction curve, Hosmer-Lemeshow test, integrated discrimination improvement (IDI), net reclassification improvement (NRI), and decision-curve analysis (DCA). The eight features with a minimal BIC model were mannitol use, mechanical ventilation, vasopressor use, international normalized ratio, urea nitrogen, respiratory rate, and cerebrovascular disease. The proposed nomogram (TBI-IHM model) was the best mortality prediction model, with better discrimination and superior model fitting for severely ill TBI patients staying in ICU. The model's receiver operating characteristic curve (ROC) was the best compared to the seven other models. It might be clinically helpful for doctors to make clinical decisions. The proposed nomogram (TBI-IHM model) has significant potential as a clinical utility in predicting mortality in TBI patients.

Combinatorial necessary conditions for regular graphs to induce periodic quantum walks

We derive combinatorial necessary conditions for discrete-time quantum walks defined by regular mixed graphs to be periodic. One useful necessary condition is that if a k-regular mixed graph with n vertices is periodic, then 2n/k must be an integer. As an application of this work, we determine periodicity of mixed complete graphs and mixed graphs with a prime number of vertices. Furthermore, we study periodicity of mixed strongly regular graphs and several classes of mixed distance-regular graphs, and extend existing results to mixed graphs.

On the multiplicities of digraph eigenvalues

We show various upper bounds for the order of a digraph (or a mixed graph) whose Hermitian adjacency matrix has an eigenspace of prescribed codimension. In particular, this generalizes the so-called absolute bound for (simple) graphs first shown by Delsarte, Goethals, and Seidel (1977) and extended by Bell and Rowlinson (2003). In doing so, we also adapt the Blokhuis' theory (1983) of harmonic analysis in real hyperbolic spaces to that in complex hyperbolic spaces.

The complexity of 2-vertex-connected orientation in mixed graphs

We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of Bang-Jensen, Huang and Zhu. For the second part, we call a directed graph D=(V,A)2T-connected for some T⊆V if D is 2-arc-connected and D−v is strongly connected for all v∈T. We deduce a characterization of the graphs admitting a 2T-connected orientation from the theorem of Thomassen.

Relation between the row left rank of a quaternion unit gain graph and the rank of its underlying graph

Let $\Phi=(G,U(\mathbb{Q}),\varphi)$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph), where $G$ is the underlying graph of $\Phi$, $U(\mathbb{Q})=\{z\in \mathbb{Q}: |z|=1\}$ is the circle group, and $\varphi:\overrightarrow{E}\rightarrow U(\mathbb{Q})$ is the gain function such that $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$. Let $A(\Phi)$ be the adjacency matrix of $\Phi$ and $r(\Phi)$ be the row left rank of $\Phi$. In this paper, we prove that $-2c(G)\leq r(\Phi)-r(G)\leq 2c(G)$, where $r(G)$ and $c(G)$ are the rank and the dimension of cycle space of $G$, respectively. All corresponding extremal graphs are characterized. The results will generalize the corresponding results of signed graphs (Lu et al. [20] and Wang [33]), mixed graphs (Chen et al. [7]), and complex unit gain graphs (Lu et al. [21]).

Moore mixed graphs from Cayley graphs

A Moore ( r , z , k ) -mixed graph G has every vertex with undirected degree r , directed in- and out-degree z , diameter k , and number of vertices (or order) attaining the corresponding Moore bound M ( r , z , k ) for mixed graphs. When the order of G is close to M ( r , z , k ) vertices, we refer to it as an almost Moore graph. The first part of this paper is a survey about known Moore (and almost Moore) general mixed graphs that turn out to be Cayley graphs. Then, in the second part of the paper, we give new results on the bipartite case. First, we show that Moore bipartite mixed graphs with diameter three are distance-regular, and their spectra are fully characterized. In particular, an infinity family of Moore bipartite (1, z , 3) -mixed graphs is presented, which are Cayley graphs of semidirect products of groups. Our study is based on the line digraph technique, and on some results about when the line digraph of a Cayley digraph is again a Cayley digraph.

English

In the study of a graph theory, degrees of vertices and number of edges are among the important theoretical terms that are helpful in showing discrete structures and their properties. This study attempts to present the relationship between number of edges and degrees of vertices in a mixed graph. Key words: Graph, directed graphs, undirected graphs, mixed graph, degree of vertex, number of edges.

HS-integral and Eisenstein integral normal mixed Cayley graphs

A mixed graph is said to be second kind hermitian integral (HS-integral) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called Eisenstein integral if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set S for which the normal mixed Cayley graph Cay(Γ,S) is HS-integral for any finite group Γ. We further show that a normal mixed Cayley graph is HS-integral if and only if it is Eisenstein integral. This paper generalizes the results of Kadyan and Bhattacharjy (2022) [11].

A universal framework for single-cell multi-omics data integration with graph convolutional networks.

Single-cell omics data are growing at an unprecedented rate, whereas effective integration of them remains challenging due to different sequencing methods, quality, and expression pattern of each omics data. In this study, we propose a universal framework for the integration of single-cell multi-omics data based on graph convolutional network (GCN-SC). Among the multiple single-cell data, GCN-SC usually selects one data with the largest number of cells as the reference and the rest as the query dataset. It utilizes mutual nearest neighbor algorithm to identify cell-pairs, which provide connections between cells both within and across the reference and query datasets. A GCN algorithm further takes the mixed graph constructed from these cell-pairs to adjust count matrices from the query datasets. Finally, dimension reduction is performed by using non-negative matrix factorization before visualization. By applying GCN-SC on six datasets, we show that GCN-SC can effectively integrate sequencing data from multiple single-cell sequencing technologies, species or different omics, which outperforms the state-of-the-art methods, including Seurat, LIGER, GLUER and Pamona.

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].