Search Results (24)

Tree-width and path-width are well-known graph parameters. Many NP-hard graph problems admit polynomial-time solutions when restricted to graphs of bounded tree-width or bounded path-width. In this work, we study the behavior of tree-width and path-width under various unary and binary graph transformations. For considered transformations, we provide upper and lower bounds for the tree-width and path-width of the resulting graph in terms of those of the initial graphs or argue why such bounds are impossible to specify. Among the studied unary transformations are vertex addition, vertex deletion, edge addition, edge deletion, subgraphs, vertex identification, edge contraction, edge subdivision, minors, powers of graphs, line graphs, edge complements, local complements, Seidel switching, and Seidel complementation. Among the studied binary transformations, we consider the disjoint union, join, union, substitution, graph product, 1-sum, and corona of two graphs. Full article

The Hengduan Mountains region (HMR) is vulnerable to flash flood disasters, which account for the largest proportion of flood-related fatalities in China. Flash flood regionalization, which divides a region into homogeneous subdivisions based on flash flood-inducing factors, provides insights for the spatial distribution patterns of flash flood risk, especially in ungauged areas. However, existing methods for flash flood regionalization have not fully reflected the spatial topology structure of the inputted geographical data. To address this issue, this study proposed a novel framework combining a state-of-the-art unsupervised Graph Neural Network (GNN) method, Dink-Net, and Shapley Additive exPlanations (SHAP) for flash flood regionalization in the HMR. A comprehensive dataset of flash flood inducing factors was first established, covering geomorphology, climate, meteorology, hydrology, and surface conditions. The performances of two classic machine learning methods (K-means and Self-organizing feature map) and three GNN methods (Deep Graph Infomax (DGI), Deep Modularity Networks (DMoN), and Dilation shrink Network (Dink-Net)) were compared for flash-flood regionalization, and the Dink-Net model outperformed the others. The SHAP model was then applied to quantify the impact of all the inducing factors on the regionalization results by Dink-Net. The newly developed framework captured the spatial interactions of the inducing factors and characterized the spatial distribution patterns of the factors. The unsupervised Dink-Net model allowed the framework to be independent from historical flash flood data, which would facilitate its application in ungauged mountainous areas. The impact analysis highlights the significant positive influence of extreme rainfall on flash floods across the entire HMR. The pronounced positive impact of soil moisture and saturated hydraulic conductivity in the areas with a concentration of historical flash flood events, together with the positive impact of topography (elevation) in the transition zone from the Qinghai–Tibet Plateau to the Sichuan Basin, have also been revealed. The results of this study provide technical support and a scientific basis for flood control and disaster reduction measures in mountain areas according to local inducing conditions. Full article

In this paper, motivated by the recently introduced topological indices—the Sombor index, Sombor coindex, and Lanzhou index, we define a new index—the Lanzhou coindex of a graph. Furthermore, we investigate the Sombor index (coindex) and the Lanzhou index (coindex) of tadpole graphs, wheel graphs, and two-dimensional grid graphs, as well as their paraline graphs. Full article

Mesh subdivision is a common mesh-processing algorithm used to improve model accuracy and surface smoothness. Its classical scheme adopts a fixed linear vertex update strategy and is implemented iteratively, which often results in excessive mesh smoothness. In recent years, a nonlinear subdivision method that uses neural network methods, called neural subdivision (NS), has been proposed. However, as a new scheme, its application scope and the effect of its algorithm need to be improved. To solve the above problems, a graph neural network method based on neural subdivision was used to realize mesh subdivision. Unlike fixed half-flap structures, the non-fixed mesh patches used in this paper naturally expressed the interior and boundary of a mesh and learned its spatial and topological features. The tensor voting strategy was used to replace the half-flap spatial transformation method of neural subdivision to ensure the translation, rotation, and scaling invariance of the algorithm. Dynamic graph convolution was introduced to learn the global features of the mesh in the way of stacking, so as to improve the subdivision effect of the network on the extreme input mesh. In addition, vertex neighborhood information was added to the training data to improve the robustness of the subdivision network. The experimental results show that the proposed algorithm achieved a good subdivision of both the general input mesh and extreme input mesh. In addition, it effectively subdivided mesh boundaries. In particular, using the general input mesh, the algorithm in this paper was compared to neural subdivision through quantitative experiments. The proposed method reduced the Hausdorff distance and the mean surface distance by 27.53% and 43.01%, respectively. Full article

The complexity (number of spanning trees) in a finite graph $\Gamma$ (network) is crucial. The quantity of spanning trees is a fundamental indicator for assessing the dependability of a network. The best and most dependable network is the one with the most spanning trees. In graph theory, one constantly strives to create novel structures from existing ones. The super subdivision operation produces more complicated networks, and the matrices of these networks can be divided into block matrices. Using methods from linear algebra and the characteristics of block matrices, we derive explicit formulas for determining the complexity of the super subdivision of a certain family of graphs, including the cycle $C_n$, where $n=3,4,5,6$; the dumbbell graph $D{b}_{m,n}$; the dragon graph $P_m\left(C_n\right)$; the prism graph ${\Pi }_n$, where $n=3,4$; the cycle $C_n$ with a $P_{\frac{n}{2}}$-chord, where $n=4,6$; and the complete graph $K_4$. Additionally, 3D plots that were created using our results serve as illustrations. Full article

Circulant networks are a very important and widely studied class of graphs due to their interesting and diverse applications in networking, facility location problems, and their symmetric properties. The structure of the graph ensures that it is symmetric about any line that cuts the graph into two equal parts. Due to this symmetric behavior, the resolvability of these graph becomes interning. Subdividing an edge means inserting a new vertex on the edge that divides it into two edges. The subdivision graph G is a graph formed by a series of edge subdivisions. In a graph, a resolving set is a set that uniquely identifies each vertex of the graph by its distance from the other vertices. A metric basis is a resolving set of minimum cardinality, and the number of elements in the metric basis is referred to as the metric dimension. This paper determines the minimum resolving set for the graphs $H_l[1,k]$ constructed from the circulant graph $C_l[1,k]$ by subdividing its edges. We also proved that, for $k=2,3$, this graph class has a constant metric dimension. Full article

In the modern era, mathematical modeling consisting of graph theoretic parameters or invariants applied to solve the problems existing in various disciplines of physical sciences like computer sciences, physics, and chemistry. Topological indices (TIs) are one of the graph invariants which are frequently used to identify the different physicochemical and structural properties of molecular graphs. Wiener index is the first distance-based TI that is used to compute the boiling points of the paraffine. For a graph F, the recently developed Gutman Connection (GC) index is defined on all the unordered pairs of vertices as the sum of the multiplications of the connection numbers and the distance between them. In this note, the $GC$ index of the operation-based symmetric networks called by first derived graph $D_1\left(F\right)$ (subdivision graph), second derived graph $D_2\left(F\right)$ (vertex-semitotal graph), third derived graph $D_3\left(F\right)$ (edge-semitotal graph) and fourth derived graph $D_4\left(F\right)$ (total graph) are computed in their general expressions consisting of various TIs of the parent graph F, where these operation-based symmetric graphs are obtained by applying the operations of subdivision, vertex semitotal, edge semitotal and the total on the graph F respectively. Full article

Entropy is essential. Entropy is a measure of a system’s molecular disorder or unpredictability, since work is produced by organized molecular motion. Entropy theory offers a profound understanding of the direction of spontaneous change for many commonplace events. A formal definition of a random graph exists. It deals with relational data’s probabilistic and structural properties. The lower-order distribution of an ensemble of attributed graphs may be used to describe the ensemble by considering it to be the results of a random graph. Shannon’s entropy metric is applied to represent a random graph’s variability. A structural or physicochemical characteristic of a molecule or component of a molecule is known as a molecular descriptor. A mathematical correlation between a chemical’s quantitative molecular descriptors and its toxicological endpoint is known as a QSAR model for predictive toxicology. Numerous physicochemical, toxicological, and pharmacological characteristics of chemical substances help to foretell their type and mode of action. Topological indices were developed some 150 years ago as an alternative to the Herculean, and arduous testing is needed to examine these features. This article uses various computational and mathematical techniques to calculate atom–bond connectivity entropy, atom–bond sum connectivity entropy, the newly defined Albertson entropy using the Albertson index, and the IRM entropy using the IRM index. We use the subdivision and line graph of the $H_{3}B{O}_3$ layer structure, which contains one boron atom and three oxygen atoms to form the chemical boric acid. Full article

The space-mapping method provides a novel method for dimension formulae explanation and basis construction for the spline space over hierarchical T-meshes. By the space-mapping method, we provide a unique basis construction framework that incorporates basis modification of the spline space over modified hierarchical T-meshes. The subdivision rules on the modified hierarchical T-meshes are given to prevent the redundant edges that exist on hierarchical T-meshes. In the basis construction framework, we describe the spline-modification mechanism over the modified hierarchical T-mesh when the cells of the corresponding crossing vertex relationship graph (CVR graph) are adjusted. We provide the framework’s algorithms for basis construction and modification. Moreover, we discuss the application of the splines that are constructed by the framework to surface reconstruction with adaptive refinement. In comparison to splines over hierarchical T-meshes, the modified hierarchical T-meshes have fewer cells subdivided when achieving similar accuracy. Full article

An irregularity index $IR\left(\mathsf{\Gamma }\right)$ of a graph $\mathsf{\Gamma }$ is a nonnegative numeric quantity (i.e., $IR\left(\mathsf{\Gamma }\right)\ge 0$) such that $IR\left(\mathsf{\Gamma }\right)=0$ iff $\mathsf{\Gamma }$ is a regular graph. In this paper, we show that $IRC$ closely correlates with the normal boiling point $T_{bp}$ and the standard heat of formation $\Delta H_f^o$ of lower benzenoid hydrocarbons. The correlation models that fit the data efficiently for both $T_{bp}$ and $\Delta H_f^o$ are linear. We develop further mathematical properties of $IRC$ by calculating its exact expressions for the recently introduced transformation graphs as well as certain derived graphs, such as the total graph, semi-total point graph, subdivision graph, semi-total line graph, double, strong double, and extended double cover graphs. Some open problems are proposed for further research on the $IRC$ index of graphs. Full article

The locating-chromatic number of a graph combines two graph concepts, namely coloring vertices and partition dimension of a graph. The locating-chromatic number is the smallest k such that G has a locating k-coloring, denoted by ${\chi }_L\left(G\right)$. This article proposes a procedure for obtaining a locating-chromatic number for an origami graph and its subdivision (one vertex on an outer edge) through two theorems with proofs. Full article

A paired-dominating set of a graph G without isolated vertices is a dominating set of vertices whose induced subgraph has perfect matching. The minimum cardinality of a paired-dominating set of G is called the paired-domination number γ_pr(G) of G. The paired-domination subdivision number sd_γpr(G) of G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. Here, we show that, for each tree T ≠ P₅ of order n ≥ 3 and each edge e ∉ E(T), sd_γpr(T) + sd_γpr(T + e) ≤ n + 2. Full article

In order to increase the paired-domination number of a graph G, the minimum number of edges that must be subdivided (where each edge in G can be subdivided no more than once) is called the paired-domination subdivision number ${\mathrm{sd}}_{{\gamma }_{pr}}\left(G\right)$ of G. It is well known that ${\mathrm{sd}}_{{\gamma }_{pr}}(G+e)$ can be smaller or larger than ${\mathrm{sd}}_{{\gamma }_{pr}}\left(G\right)$ for some edge $e\notin E\left(G\right).$ In this note, we show that, if G is an isolated-free graph different from $m{K}_2,$ then, for every edge $e\notin E\left(G\right)$, ${\mathrm{sd}}_{{\gamma }_{pr}}(G+e)\le {\mathrm{sd}}_{{\gamma }_{pr}}\left(G\right)+2\Delta \left(G\right)$. Full article

For a graph G with no isolated vertex, let ${\gamma }_{pr}\left(G\right)$ and ${\mathrm{sd}}_{{\gamma }_{pr}}\left(G\right)$ denote the paired-domination and paired-domination subdivision numbers, respectively. In this note, we show that if T is a tree of order $n\ge 4$ different from a healthy spider (subdivided star), then ${\mathrm{sd}}_{{\gamma }_{pr}}\left(T\right)\le min\{\frac{{\gamma }_{pr}\left(T\right)}{2}+1,\frac{n}{2}\},$ improving the $(n-1)$-upper bound that was recently proven. Full article

The precise application of tightening torque is one of the important measures to ensure accurate bolt connection and improvement in product assembly quality. Currently, due to the limited assembly space and efficiency, a wrench without the function of torque measurement is still an extensively used assembly tool. Therefore, wrench torque monitoring is one of the urgent problems that needs to be solved. This study proposes a multi-segmentation parallel convolution neural network (MSP-CNN) model for estimating assembly torque using surface electromyography (sEMG) signals, which is a method of torque monitoring through classification methods. The MSP-CNN model contains two independent CNN models with different or offset torque granularities, and their outputs are fused to obtain a finer classification granularity, thus improving the accuracy of torque estimation. First, a bolt tightening test bench is established to collect sEMG signals and tightening torque signals generated when the operator tightens various bolts using a wrench. Second, the sEMG and torque signals are preprocessed to generate the sEMG signal graphs. The range of the torque transducer is divided into several equal subdivision ranges according to different or offset granularities, and each subdivision range is used as a torque label for each torque signal. Then, the training set, verification set, and test set are established for torque monitoring to train the MSP-CNN model. The effects of different signal preprocessing methods, torque subdivision granularities, and pooling methods on the recognition accuracy and torque monitoring accuracy of a single CNN network are compared experimentally. The results show that compared to maximum pooling, average pooling can improve the accuracy of CNN torque classification and recognition. Moreover, the MSP-CNN model can improve the accuracy of torque monitoring as well as solve the problems of non-convergence and slow convergence of independent CNN network models. Full article