ABSTRACT For an integer is the smallest value such that any 2‐edge connected mixed graph with a radius has an orientation with radius of no more than . In a recent significantly advancement, Babu, Benson, and Rajendraprasad showed that is at most . This finding is detailed in their paper “Improved bounds for the oriented radius of mixed multigraphs,” published in J. Graph Theory , 103 (2023), 674‐689. Furthermore, they proved that has an orientation with a radius of at most if every belongs to a cycle of length at most . The authors' results were derived through Observation 1, which served as the foundation for the development of Algorithm ORIENTOUT and Algorithm ORIENTIN. By integrating these algorithms, they obtained the improved bounds. However, an error has been identified in Observation 1, necessitating revisions to Algorithm ORIENTOUT and Algorithm ORIENTIN. In this note, we address the error and propose the necessary modifications to both algorithms, thereby ensuring the correctness of the conclusions.

Causal graphs provide a rigorous framework for encoding causal assumptions. Yet, their integration with multilevel models remains limited. We review common path diagram conventions in multilevel modeling and show, through a bivariate regression example, that these function as statistical model visualizations rather than causal graphs. We formalize parametric cross-sectional multilevel models as parametric structural causal models with linear causal effects (conditional on possible moderators) and Gaussian error terms. We then translate them into acyclic directed mixed graphs. We illustrate this framework using a well-known empirical example, the High School and Beyond study. This work provides a systematic bridge between cross-sectional parametric multilevel models and modern causal graph theory.

We characterize twisted right-angled Artin groups (T-RAAGs) that are subgroup separable using only their defining mixed graphs: such a group is subgroup separable if and only if the underlying simplicial graph contains no induced path or square on four vertices. This generalizes the results of Metaftsis and Raptis on classical right-angled Artin groups. We also show that the subgroup membership problem is decidable when the group is coherent—which occurs precisely when the defining mixed graph is chordal. Furthermore, we exhibit a family of cone-type mixed graphs for which the corresponding T-RAAGs have decidable rational and submonoid membership problems.

ABSTRACT We study the entanglement properties of randomized mixed hypergraph states, extending the concept of randomized mixed graph states to encompass hypergraph‐based quantum states. In our model, imperfect generalized multi‐qubit gates are applied probabilistically, simulating experimentally realistic noisy gate operations where gate fidelity decreases with increasing hyperedge order. We analyze bipartite and genuine multipartite entanglement of these mixed multi‐qubit states. Numerical results for various hypergraph configurations with up to four qubits reveal rich, sometimes nonmonotonic entanglement behavior stemming from the interplay between hyperedge structure and gate imperfections. We derive analytical expressions for entanglement witnesses based on randomization overlap for new hypergraph families. Our findings contribute to understanding entanglement resilience under gate imperfections, providing insight into the experimental implementation of hypergraph states in noisy quantum devices.

Diameter two orientability of mixed graphs

Multi-agent trajectory prediction with trend-aware attention and mixed graph convolutional networks

In evolutionary biology, phylogenetic networks are graphs that provide a flexible framework for representing complex evolutionary histories that involve reticulate evolutionary events. Recently, phylogenetic studies have started to focus on a special class of such networks called semi-directed networks. These graphs are defined as mixed graphs that can be obtained by de-orienting some of the arcs in some rooted phylogenetic network, that is, a directed acyclic graph whose leaves correspond to a collection of species and that has a single source or root vertex. However, this definition of semi-directed networks is implicit in nature since it is not clear when a mixed-graph enjoys this property or not. In this paper, we introduce novel, explicit mathematical characterizations of semi-directed networks, and also multi-semi-directed networks, that is mixed graphs that can be obtained from directed phylogenetic networks that may have more than one root. In addition, through extending foundational tools from the theory of rooted networks into the semi-directed setting—such as cherry picking sequences, omnians, and path partitions—we characterize when a (multi-)semi-directed network can be obtained by de-orienting some rooted network that is contained in one of the well-known classes of tree-child, orchard, tree-based or forest-based networks. These results address structural aspects of (multi-)semi-directed networks and pave the way to improved theoretical and computational analyses of such networks, for example, within the development of algebraic evolutionary models that are based on such networks.

We consider Hoffman's program about the limit points of the spectral radius of the Hermitian adjacency matrix of mixed graphs. In particular, we determine all such limit points. As an intermediate step, we determine all mixed graphs without negative 4-cycle whose spectral radius does not exceed 2+5.

Abstract An ( r , z ; g )-mixed graph is a graph containing both edges and darts satisfying the regularity property that each vertex of the graph is incident to r edges, z ingoing and z outgoing darts (called total regularity), and being of oriented girth g , i.e., containing an oriented cycle of length g , and no shorter oriented cycles. The problem addressed in this paper is analogous to the Cage Problem and calls for determining the orders of the smallest totally regular ( r , z ; g )-mixed graphs. We derive several upper and lower bounds on the orders of such minimal graphs, study the relations between these extremal graphs and their non-oriented or digraphical counterparts, and focus on properties of totally regular mixed graphs obtained by replacing some of the edges of the incidence graphs of projective and biaffine planes by darts. We also introduce two constructions based on introducing additional edges or darts into induced subgraphs of these incidence graphs.

Spectral Analysis of Normalized Hermitian Laplacian Matrices in Random Mixed Graphs

Relation between the H-rank of a mixed graph and the girth of its underlying graph

• A novel Mixed Graph Construction tailored for cooperative machinery dynamics, enabling comprehensive relationships. • Improved model reliability with robustness when handling missing data. • Inclusion of cooperating machinery for improved activity classification performance. • Enhanced applicability of computer vision-based approaches in construction monitoring. Existing computer vision-based approaches struggle to identify machinery actions due to the challenges posed by environmental complexity and various obstructions in construction. This study introduces a novel two-stage framework that benefits from a newly proposed Residual Fusion Graph Convolution Network (RFGCN) to classify machinery actions with enhanced robustness and accuracy. The framework first extracts key machinery components from video data, subsequently transforming them into a graph-based representation. This spatio-temporal graph is then fed into the RFGCN model, specifically designed to overcome issues like partial obstructions and missing information common in busy construction sites. Experimental evaluations reveal the method’s high efficacy, achieving an accuracy of up to 96.4% and outperforming state-of-the-art. Additionally, the proposed RFGCN model achieved state-of-the-art performance on four established benchmark datasets for graph classification using spatial data only. These results suggest the potential of the proposed framework in facilitating the transition towards more intelligent and automated construction sites.

Mixed graph neural network for session-based recommendation with category-aware noise filtering

Let G be a multidigraph without self-loops. The complex Laplacian matrix of G, denoted by L C ( G ) , is defined in Barik et al. [On singularity and properties of eigenvectors of complex Laplacian matrix of multidigraphs. AKCE Int J Graphs Comb 2023;20(2):125–133. doi: 10.1080/09728600.2023.2234014]. In this article, we consider the complex signless Laplacian matrix of multidigraphs, denoted by Q C ( G ) . For a simple graph, the Matrix-Tree Theorem gives the number of spanning trees in a graph in terms of the principal minors of its Laplacian matrix. That gives a motivation to study the principal minors of the matrices, which has been done for simple graphs, digraphs, signed graphs and mixed graphs. In this article, we provide a combinatorial description of the principal minors and determinants of both L C ( G ) and Q C ( G ) . As an application, a class of multidigraphs is provided whose complex Laplacian spectrum and complex signless Laplacian spectrum are the same. We obtain a necessary and sufficient condition for a multidigraph to be Q C -singular. Further, the eigenvectors of the Q C -singular multidigraphs are studied.

Group theory and graph theory have important research value in mathematics today. Counting problems also play a decisive role in combinatorics. This paper introduces the number of isomorphism classes of mixed graphs with n vertices. The counting of them which using Burnside’s lemma is solved by converting the cases of edges and vertices to some colors.

An upper bound for the largest singular value of extended mixed graphs

The CD(n,q) graphs are connected components of q-regular graphs D(n,q) introduced in 1995 by Lazebnik and Ustimenko. They constitute the best universal family of regular graphs of prime power degree with regard to the Cage Problem which calls for determining the orders of the smallest k-regular graphs of girth g. The girths of the CD(n,q) graphs are known to be at least $n+4$ in case of even n, and n+5 for odd n. We propose to extend the use of the CD(n,q) graphs into the area of mixed graphs by adding directions to certain edges of the C(n,q)graphs.In the context of mixed graphs, graphs in which the number of incident non-oriented edges is the same for all vertices, and the numbers of out-going and in-going edges are also equal and the same for all vertices, are of special interest and are called totally regular mixed graphs. In view of the special properties of the original C(n,q) graphs with regard to cages, we believe that the totally regular mixed graphs we propose to study may also prove to be extremal with regard to properties sought for in the area of mixed graphs.

Graph neural networks (GNNs) have gained superior performance in graph-based prediction tasks with a variety of applications such as social analysis and drug discovery. Despite the remarkable progress, their performance often degrades on test graphs with distribution shifts. Existing domain adaptation methods rely on unlabeled test graphs during optimization, limiting their applicability to graphs in the wild. Towards this end, this paper studies the problem of multi-domain generalization on graphs, which utilizes multiple source graphs to learn a GNN with high performance on unseen target graphs. We propose a new approach named Topological Adversarial Learning with Prototypical Mixup (TRACI) to solve the problem. The fundamental principle behind our TRACI is to produce virtual adversarial and mixed graph samples from a data-centric view. In particular, TRACI enhances GNN generalization by employing a gradient-ascent strategy that considers both label prediction entropy and graph topology to craft challenging adversarial samples. Additionally, it generates domain-agnostic node representations by characterizing class-graph pair prototypes through latent distributions and applying multi-sample prototypical Mixup for distribution alignment across graphs. We further provide theoretical analysis showing that TRACI reduces the model's excess risk. Extensive experiments on various benchmark datasets demonstrate that TRACI outperforms state-of-the-art baselines, validating its effectiveness.

A mixed graph is a graph obtained from a simple undirected graph by orientating a subset of edges. In 2020, Mohar introduced a new kind of Hermitian adjacency matrix (called Eisenstein adjacency matrix) of a mixed graph using a primitive sixth root of unity, which has some advantages over the one proposed by Guo and Mohar in 2017, and independently by Liu and Li in 2015 (called Gaussian adjacency matrix). We consider the problem of generalized spectral characterizations of mixed graphs based on the Eisenstein adjacency matrix. A simple sufficient condition is given for a self-converse mixed graph to be determined by its generalized Eisenstein spectrum based on the ring of Eisenstein integers. Numerical experiments are also presented which show that the generalized Eisenstein spectrum is superior to the generalized Gaussian spectrum in distinguishing mixed graphs.

A PM2.5 spatiotemporal prediction model based on mixed graph convolutional GRU and self-attention network.