The maximum number of maximum dissociation sets in potted graphs

The minimum number of maximal dissociation sets in unicyclic graphs

Real-world systems frequently exhibit hierarchical multipartite graph structures, yet existing graph neural network (GNN) approaches lack systematic frameworks for hyperparameter optimization in heterogeneous multi-level architectures, limiting their practical applicability. This study proposes a Bayesian optimization framework specifically designed for heterogeneous GNNs operating on three-level graph structures, addressing the computational challenges of configuring partition-aware architecture. Four GNN architectures—Graph Convolutional Networks (GCNs), Graph Attention Networks (GATs), Graph Isomorphism Networks (GINs), and GraphSAGE—were systematically evaluated using Gaussian Process-based Bayesian hyperparameter optimization with inter-partition message-passing mechanisms. The framework was validated on the TIMSS 2023 dataset (10,000 students, 789 schools, 25 countries), demonstrating that Bayesian-optimized GraphSAGE achieved the highest explained variance (R2 = 0.6187, RMSE = 71.73, MAE = 64.32) compared to seven baseline methods. Bayesian optimization substantially improved model performance, revealing that two-layer architectures optimally capture cross-partition dependencies in three-level structures. GNNExplainer was used to identify the most influential student-level features learned by the model, providing explanatory insight into how the model represents individual characteristics. The optimization framework is broadly applicable to other heterogeneous and multi-level graph settings; however, the empirical findings, such as the optimal architecture depth, are specific to hierarchical graphs with structural properties like the TIMSS topology.

This article studies the minimum rank of a (simple, undirected) graph, which is the minimum rank among all matrices in a space determined by the graph. It determines the exact set of graphs on eight vertices for which the nullity of a minimum rank matrix does not coincide with a bound determined by the zero forcing number of a graph. Although the goal was to determine which eight-vertex graphs satisfy maximum nullity equal to the zero forcing number, it also establishes several additional methods to assist in the computation of minimum rank for general graphs.

Monitoring edge-geodetic sets in graphs

On perfect dominating sets in Cayley graphs

Traditionally, the quality-of-service assessment of a roundabout is only focused on the capacity of the individual entries. However, the exits also have an important impact on the performance of a roundabout since they can be temporarily blocked by prioritized crossing pedestrians. As a consequence, exiting vehicles may spill back from the blocked pedestrian crossing onto the circular roadway. At single-lane roundabouts this will cause a queue on the circle, which again can block the upstream entries and reduce their capacity. The extent of this effect is influenced by vehicle traffic volumes, pedestrian volumes, and also the geometric design of the specific roundabout. In practice, an analytical method would be desirable to estimate the effects of this phenomenon. This paper presents such a solution. As a starting point a mathematical model based on queuing theory is developed which calculates the probability of an entry to the roundabout being obstructed by a queue originating from the next exit downstream. This probability is applied to reduce the usual estimated capacity of the entry. The theoretical model is compared with microscopic simulations of a single-lane roundabout. As a result, a high correspondence between the theoretical approach and the simulation, along with a measurement at a real-world roundabout, becomes evident. Therefore, the model is recommended for practical application in, for example, the Highway Capacity Manual (HCM) or the German Highway Capacity Manual (HBS). A set of graphs is presented which can be introduced into such a standardized calculation procedure.

In this article two new sets associated with the vertex set of a finite graph namely equitable degree set and non-equitable degree set are derived. Furthermore, equitable and non-equitable number of some standard classes of graphs identity graph is evaluated.

We study the problem of partitioning the edge set of the complete graph into bipartite subgraphs under certain constraints defined by forbidden subgraphs. These constraints lead to both classical problems, such as partitioning into independent matchings or complete bipartite subgraphs, and novel variants motivated by structural restrictions. Our theoretical framework is inspired by clustering problems in real-world transaction graphs, which can be formulated naturally as edge partitioning problems under bipartite graph constraints. The main result of this paper is the proof of the bounds for $χ'_{2K_2}(n)$, which corresponds to the minimum number of induced $2K_2$-free bipartite subgraphs needed to partition the edges of $K_n$. In addition to this central result, we also present several similar bounds for other forbidden subgraphs on three or four vertices. Some are included primarily for the sake of completeness, to demonstrate the broad applicability of our approach, and some lead to other novel or well-known graph theoretical problems.

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$ that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$ consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. Let $\partial$ denote the path-length distance function of $J_q(n,k)$. Pick vertices $x,y$ of $J_q(n,k)$ such that $1<\partial(x,y)<k$. Let $\text{Stab}(x,y)$ denote the subgroup of $GL(V)$ that stabilizes both $x$ and $y$. In this paper, we investigate the orbits of $\text{Stab}(x,y)$ acting on the local graph $\Gamma(x)$. We show that there are five orbits. By construction, these five orbits give an equitable partition of $\Gamma(x)$; we find the corresponding structure constants. In order to describe the five orbits more deeply, we bring in a Euclidean representation of $J_q(n,k)$ associated with the second largest eigenvalue of $J_q(n,k)$. By construction, for each orbit its characteristic vector is represented by a vector in the associated Euclidean space. We compute many inner products and linear dependencies involving the five representing vectors.

A set $S$ of vertices of a graph $G$ is a strong (resp. weak) dominating set of $G$ if for every vertex $v$ of $G$ outside of $S$, there is a vertex $u$ inside of $S$ such that $u$ and $v$ are adjacent and $deg_G(v)\le deg_G(u)$ (resp. $deg_G(v)\ge deg_G(u)$). The minimum cardinality of a strong (resp. weak) dominating set is called the strong (resp. weak) domination number of $G$, and is denoted by $\gamma_s(G)$ (resp. $\gamma_w(G)$). In this paper, we characterize the strong and weak dominating sets of graphs under some binary operations. As a result, we also determine the exact values of or sharp bounds for the corresponding strong and weak domination numbers.

Two single-crystal X-ray structure determinations of 2-aminopyrimidinium hydrogen tri oxofluorophosphate, (C4H6N3)+·(HFO3P)−, (I), and bis(2-aminopyrimidinium) trioxofluorophosphate, 2(C4H6N3)+·(FO3P)2−, (II), as well as their vibration spectra (FTIR on powder samples and the Raman spectra on unoriented single crystals) with a detailed assignment of vibrational modes are reported. The structure (I) consists of one independent 2-aminopyrimidinium cation and one hydrogen trioxofluorophosphate anion, while (II) consists of two symmetry independent 2-aminopyrimidinium cations and one trioxofluorophosphate anion. In (I), there is an O-H···O hydrogen bond of a moderate strength. A pair of these hydrogen bonds is situated about the symmetry centre and involved in the graph set motif R22(8). There are also N-H···O hydrogen bonds of a moderate strength, which are present in both structures while being involved in the graph set motifs R22(8), too. In addition, the N-H···O hydrogen bonds form R34(10) graph set motifs in (II). The latter motifs form ribbons which propagate parallel to the unit-cell axis a. In both structures, there are present π···π-electron ring interactions into which the primary amine groups are involved. In both structures, there are also present weak C-H···N hydrogen bonds with participation of the non-protonated ring N-atoms. The fluorine participates in the C-H···F hydrogen bonds in both title structures. The P-F distances are normal in both anions. The structure (I) differs from the known structure of 2-aminopyrimidinium hydrogen phosphite, the compositional isomer, though the main hydrogen bonds show similar geometry in both structures. The crystal of (I) was twinned.

Finding a Minimum Source Set in Temporal Graphs

In a distributed system framework, spatial crowdsourcing (SC) is a highly important area of research where task allocation to task executors (TEs) is an important step. Tasks are requested by a task provider and are allocated by an SC platform to TEs. However, TEs may submit the allocated task as late as possible, known as procrastination. Plenty of research works are available on task allocation in SC, whereas few research works are found that address procrastination. In a bipartite graph setting, a procrastination-aware scheduling is proposed. A recent work uses ChatGPT for procrastinating agents. Balanced distribution of tasks has not been addressed there. Recently, an algorithm was proposed that distributes tasks in a balanced manner in different slots to mitigate procrastination in SC. Here, we propose a quality-aware task allocation mechanism in an SC environment that combines a data science approach with a reinforcement learning-based approach. Once TEs are allocated tasks, we have proposed an AI-enabled (learning-the-variance) algorithm to distribute the tasks into slots with a more balanced distribution than any of the existing algorithms to mitigate procrastination. Our procrastination prevention mechanism outperforms existing methods, which is shown by extensive simulations. Analytically, it is shown that the proposed mechanism maintains a balanced distribution.

The research object is the process of route construction in infocommunication networks. A routing model based on a graph neural network (GNN) with edge-level classification has been developed, implemented, and experimentally studied. The model uses the GENConv architecture. Architectural features of GNNs are analyzed, including message passing, attention mechanisms, feature aggregation, and updating techniques. A comparison of GCN, GAT, and GENConv architectures is conducted, and GENConv is justified as the base architecture for edge-level classification of routing edges. A model is built using GENConv with an MLP decoder. The model is trained on a large set of graphs and evaluated in terms of accuracy, average latency, and the percentage of successfully constructed routes. The proposed approach is compared with classical algorithms based on decision quality and execution time. It is established that in inference mode, the graph model works significantly faster – especially on large graphs – and does not require exhaustive route space reprocessing for each query. This makes the proposed method suitable for real-time use in dynamic networks, where decision-making speed is critical.

Cellular automata (CAs) are used to model rule-based evolutionary systems with standard CAs applying unitary, fixed rules to an entire generation at a time. A sequential updating asynchronous cellular automaton (CA) with more than one rule for each input sequence is studied. These multiway sequential CAs (MSCAs) can model complex systems with multiple branching rule sets where changes propagate through the system. This paper examines the case of one-dimensional, two-cell, two-branch MSCAs in order to better understand their structure and the impact of parameters. The complete set of 1296 M-type rule sets possible for this type of multiway sequential CA (MSCA) is applied to a full set of 32 initial conditions, representing all possibilities of a six-cell initial condition, generating 41472 state graphs. Machine learning is used to classify a subset of these state graphs into 10 classes. Analytical data enables characterization of these classes of graphs and investigation of the role of rule sets in these state graphs. Target distribution analysis of the M-type rule sets is performed within each class of graphs to tease out intrinsic characteristics of the classes.

We design the first node-differentially private algorithm for approximating the number of connected components in a graph. Given a database representing an \( n \) -vertex graph \( G \) and a privacy parameter \(\varepsilon\) , our algorithm runs in polynomial time and, with probability \(1-o(1)\) , has additive error \(\widetilde{O}(\frac{\Delta^{*}\ln\ln n}{\varepsilon}),\) where \(\Delta^{*}\) is the smallest possible maximum degree of a spanning forest of \(G.\) Node-differentially private algorithms are known only for a small number of database analysis tasks. A major obstacle for designing such an algorithm for the number of connected components is that this graph statistic is not robust to adding one node with arbitrary connections (a change that node-differential privacy is designed to hide): every graph is a neighbor of a connected graph. We overcome this by designing a family of efficiently computable Lipschitz extensions of the number of connected components or, equivalently, the size of a spanning forest. The construction of the extensions, which is at the core of our algorithm, is based on the forest polytope of \(G.\) We prove several combinatorial facts about spanning forests, in particular, that a graph with no induced \(\Delta\) -stars has a spanning forest of degree at most \(\Delta\) . With this fact, we show that our Lipschitz extensions for the number of connected components equal the true value of the function for the largest possible monotone families of graphs. More generally, on all monotone sets of graphs, the \(\ell_{\infty}\) error of our Lipschitz extensions is nearly optimal.

Given a finite group G, let V (G) = fpep(G)jp 2 ρ(G)g, where ρ(G) is the set of prime divisors of the degrees of all irreducible characters of G and pep(G) =maxfχ(1)pjχ 2 Irr(G)g. In fact, V (G) is the vertex set of the prime-power graph of G. An interesting topic is to study if a finite simple group M can be uniquely determined by its order jMj and V (M). It has been proved that the simple groups L2(p2) and L2(p3) can be uniquely determined by its orders and vertex set of its prime-power graphs, respectively, where p is a prime. In this paper, we continue this topic and show that G ∼ = L3(p) if and only if jGj = jL3(p)j and V (G) = V (L3(p)), where p is a prime.

In this paper, we introduce the concept of Intuitionistic Fuzzy Zero-Divisor Graph (IFZDG) by incorporating Fuzzy Zero-Divisor Graph and intuitionistic fuzzy set theory. For the commutative ring [Formula: see text], let [Formula: see text] be the zero-divisor graph whose vertices are non-zero zero-divisors of [Formula: see text]. We explore an algebraic-graph-based approach to fault tolerance using Intuitionistic Fuzzy Zero-Divisor Graphs. Throughout this paper, we denote the Intuitionistic Fuzzy Zero-Divisor Graph (IFZDG) of [Formula: see text] by [Formula: see text].

The Spectral Barycentre of a Set of Graphs with Community Structure