Number Of Vertices Research Articles

Extremal spectral radii of uniform supertrees

For a hypergraph $\mathcal{G}=(V, E)$ consisting of a nonempty vertex set $V=V(\mathcal{G})$ and an edge set $E=E(\mathcal{G})$, its adjacency matrix $\mathcal {A}_{\mathcal{G}}=[(\mathcal {A}_{\mathcal{G}})_{ij}]$ is defined as $(\mathcal {A}_{\mathcal{G}})_{ij}=\sum_{e\in E_{ij}}\frac{1}{|e| - 1}$, where $E_{ij} = \{e \in E : i, j \in e\}$. The spectral radius of a hypergraph $\mathcal{G}$, denoted by $\rho(\mathcal {G})$, is the maximum modulus among all eigenvalues of $\mathcal {A}_{\mathcal{G}}$. In this paper, we represent some results on the spectral radius changing under some graphic structural perturbations. With these results, among all $k$-uniform ($k\geq 3$) supertrees with fixed number of vertices, the supertrees with the maximum, the second maximum, and the minimum spectral radius are completely determined, respectively.

GROUP MEAN CORDIAL LABELING OF SOME QUADRILATERAL SNAKE GRAPHS

Let G be a (p, q) graph and let A be a group. Let f : V (G) −→ A be a map. For each edge uv assign the label [o(f (u))+o(f (v)) / 2]. Here o(f (u)) denotes the order of f (u) as an element of the group A. Let I be the set of all integerslabeled by the edges of G. f is called a group mean cordial labeling if the following conditions hold: (1) For x, y ∈ A, |vf (x) − vf (y)| ≤ 1, where vf (x) is the number of vertices labeled with x. (2) For i, j ∈ I, |ef (i) − ef (j)| ≤ 1, where ef (i) denote the number of edges labeled with i. A graph with a group mean cordial labeling is called a group mean cordial graph. In this paper, we take A as the group of fourth roots of unity and prove that, Quadrilateral Snake, Double Quadrilateral Snake, Alternate Quadrilateral Snake and Alternate Double Quadrilateral Snake are groupmean cordial graphs.

The scaling limit of random cubic planar graphs

AbstractWe study the random cubic planar graph with an even number of vertices. We show that the Brownian map arises as Gromov–Hausdorff–Prokhorov scaling limit of as tends to infinity, after rescaling distances by for a specific constant . This is the first time a model of random graphs that are not embedded into the plane is shown to converge to the Brownian map. Our approach features a new method that allows us to relate distances on random graphs to first‐passage percolation distances on their 3‐connected core.

Analyzing the modified symmetric division deg index: mathematical bounds and chemical relevance

Abstract The modified symmetric division deg (MSD) index of a graph G is precisely defined as MSD ( G ) = ∑ j ∼ k 1 2 d j d k + d k d j , where d j and d k represent the degrees of j and k respectively. In this paper, we present precise bounds for the modified symmetric division deg index, expressed in the relation to the minimum and maximum degrees, the order and size of the graph, the number of pendant vertices and the minimum degree of a non-pendant vertex, the forgotten topological index, the modified second Zagreb index. We determine the upper bounds for the modified symmetric division deg index of unicyclic, bicyclic and k-cyclic graphs. Moreover, upon analyzing the modified symmetric division deg index, we observe its correlation with other well-known indices and its chemical applicability on the molecular graphs of octane isomers. At the end, we find the chemical applicability of the modified symmetric division deg index on benzenoid hydrocarbons and observe that the modified symmetric division deg index has a very strong correlation with the physical properties of benzenoid hydrocarbons.

Fuss’ Relation for Bicentric Tridecagon and Bicentric Pentadecagon

This article describes a general analytical derivation of the Fuss’ relation for bicentric polygons with an odd number of vertices. In particular, we derive the Fuss’ relations for the bicentric tridecagon and the bicentric pentadecagon.

A new spectral-based index for graphs and its application to polyaromatic compounds

In this study, we introduce a novel index called the modified inverse sum indeg (MISI) energy as an extension of the traditional inverse sum indeg (ISI) energy and neighbor degree sum energy. We devised an algorithm that employs the simplified molecular input line entry system (SMILES) formula of compounds to compute the MISI energy of polycyclic aromatic hydrocarbon (PAH). Additionally, we established the bounds of the MISI energy for certain classes of graphs with fixed number of vertices. A strong correlation between the MISI energy and the total [Formula: see text]-electron energy of PAHs is observed through regression analysis. Remarkably, this correlation outperforms that achieved by the classical ISI energy. These findings highlight the enhanced effectiveness of the MISI energy as a descriptor for capturing significant molecular properties. Moreover, our study establishes a foundation for further theoretical extensions and practical applications in the field of chemical graph theory.

Exploring Mean Cordial Labeling Possible Combination Position of Vertices in Graphs: A Computational Approach

Objectives: We have developed the Python module of mean cordial labeling for the different types of graphs, like Path, Cycle, and Subdivision of Star graphs. We aim to identify the number of combination positions of vertices that satisfy the mean cordial labeling rules. We have found mathematical equations that describe the labeling behavior in these graphs. Methods: In this paper, a Python program was developed to find the mean cordial labeling of the Path, Cycle, and Subdivision of the Star graph. We have obtained the number of combination positions of vertex obeying the mean cordial labeling condition using this Python module. We have drawn the graph between the number of vertices and the number of combination positions of the vertex. Findings: We have found that the different graph types like Path, Cycle, and Subdivision of Star graph satisfy the mean cordial labeling condition. The result indicates that the number of combination positions of the Path and cycle graph fulfill the power equation Y=a*xb and the Subdivision of the Star graph meets the exponential growth equation Y=a*bx. Novelty: In this paper, a new method has been developed to find the number of combination position of varies graphs using the Python module. This computational approach will be useful to analyze the biological networking and protein-protein interaction study. Keywords: Mean Cordial, Path, Cycle, Subdivision of Star graph, Computational graph theory

Efficient Rendering of Digital Twins Consisting of Both Static And Dynamic Data

Abstract. The high number of vertices and faces present in a digital twin of a large scene poses a challenge when executing large-scale simulations that require the data to be rendered multiple times with slight changes such as the light conditions in a day-night cycle during a long period of time (weeks and even months). Adding urban morphology such as tree planting introduces a new layer of complexity due to the requirement for evaluating different tree configurations and their effect on the local environment in the context of urban heat islands. Using advanced rendering techniques such as splitting a single buffer into sub-regions with inplace updates as well as buffer orphaning multiple methodologies are evaluated.

Connectivity and diagnosability of the complete Josephus cube networks under h-extra fault-tolerant model

The h-extra connectivity and the h-extra diagnosability are key parameters for evaluating the reliability and fault-tolerance of the interconnection networks of the multiprocessor systems, and play an important role in designing and maintaining interconnection networks. Recently, various self-diagnostic models have emerged to assess the fault-tolerance in interconnection networks. These interconnection networks are typically expressed by a connected graph G(V,E). For a non-complete graph G and h≥0, the h-extra cut signifies a vertex subset R of G, whose removal results in G−R disconnected, with each remaining component containing at least h+1 vertices. And the h-extra connectivity of G is defined as the minimum cardinality of all h-extra cuts of G. The h-extra diagnosability for a graph G denotes the maximum number of detectable faulty vertices when focusing on these h-extra faulty sets only. The complete Josephus cube CJCn, a variant of Qn, exhibits superior properties compared to hypercube Qn, and also boasts higher connectivity. In this study, with the help of the exact value of the h-extra connectivity of CJCn, the explicit expression of h-extra diagnosability of CJCn under both the PMC model for n≥5 and 1≤h≤⌊n−32⌋ and the MM* model for n≥5 and 2≤h≤⌊n−32⌋ are identified to share the same value (h+1)n−(h−12)+1.

Mutual-visibility and general position in double graphs and in Mycielskians

The general position problem in graphs is to find the largest possible set of vertices with the property that no three of them lie on a common shortest path. The mutual-visibility problem in graphs is to find the maximum number of vertices that can be selected such that every pair of vertices in the collection has a shortest path between them with no vertex from the collection as an internal vertex. Here, the general position problem and the mutual-visibility problem are investigated in double graphs and in Mycielskian graphs. Sharp general bounds are proved, in particular involving the total and the outer mutual-visibility number of base graphs. Several exact values are also determined, in particular the mutual-visibility number of the double graphs and of the Mycielskian of cycles.

On radio graceful Hamming graphs of any diameter

Abstract We show that Hamming graphs of diameter D are radio graceful provided that each component of the Hamming graph is a complete graph with at least D vertices. This generalizes the results known previously for diameters D = 2 and D = 3 and for general D with the assumption that the components of the Hamming graph have the same or pairwise relatively prime numbers of vertices.

Steiner 3-Wiener Index of Zigzag Polyhex Nanotubes.

Let G be a connected graph and S be a k element subset of the vertex set V(G) of G. Steiner distance is a natural generalization of the usual graph distance. The Steiner-k distance dG(S) between the vertices of S is the minimum size among all connected subgraphs whose vertex set contains S. The generalized indices based on Steiner distances have several applications in the real world. It is a well-known fact that "Steiner Problem" is NP-complete and hence any parameter based on Steiner distance is also an NP problem. The objective of this work is to determine the Steiner 3-Wiener index for an important chemical structure called the zigzag polyhex nanotube. In this paper, we present an algorithm for computing the Steiner 3-Wiener index (SW3) for zigzag polyhex nanotubes. The developed algorithm addresses the complexities associated with exponential increments in the number of vertices as the nanotube's circumference or length expands. The obtained SW3 values for zigzag polyhex nanotubes can be used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR) analyses. We have presented an algorithm and listed the numerical values of SW3 for various parameters of circumference and length of a zigzag polyhex nanotube to facilitate their utilization in QSAR/ QSPR analyses. We have obtained the time complexity for the algorithm, which shows that the SW3 values are computationally intensive. To explain this complex nature, we have used multiple linear regression to fit log SW3 values corresponding to log p and log q, the radius and length of a nanotube. The algorithm addresses the complexities associated with the exponential increase in vertices as the nanotube's circumference or length expands. Furthermore, we provide SW3 values for various parameter combinations up to circumference or length 17, and a general relation to determine the value of SW3, facilitating its utilization in real-world applications. These values serve as crucial descriptors for understanding the structural nuances of zigzag polyhex nanotubes, that find applications in material science, drug design, drug discovery, etc.

Multi-Level Sum of Product (SOP) Network Power Optimization Based on Switching Graph

The article presents the methodology of optimization of technology mapping of a multi-output function implemented in the form of sum of product (SOP) networks. The optimization is based on the concept of reducing the switching activity of combinational circuits. The aim of reducing network switching is to limit the consumption of dynamic power. Since dynamic power is one of the components of the total power consumed by digital systems, this leads to an optimization focused on the energy efficiency of digital systems. The basis of the proposed method is the technology mapping using a modified output graph describing the result of minimizing the multi-output function. The modified output graph, in terms of parameters associated with dynamic power, is defined as a switching graph. It was assumed that the key parameter associated with dynamic power is the switching activity of individual nodes of the logic network. The article presents elements of switching graph optimization which lead to the improvement of parameters associated with the dynamic power of the circuit. The essence of the proposed optimization methods is the appropriate movement of products and connections occurring in the logic network. Reducing the number of logic network vertices is also extremely important. The effectiveness of the proposed method was confirmed by the results of experiments performed on selected benchmarks. A significant reduction in the total value of switching activity was obtained for the optimized structures.

Insecticide exposure can increase burrow network production and alter burrow network structure in soil dwelling insects (Agriotes spp.)

Insecticide treated seeds are commonly used to reduce yield losses from burrowing insect damage such as wireworms. Using temporal X-ray Computed Tomography (CT) of soil-filled bioassays, we aimed to quantify changes in burrow network production and structure as a measure of wireworm behavioural change in response to three types of insecticide treated maize seed; compound X (R&D product in field trial stage of development); tefluthrin and thiamethoxam. A biopesticide alternative treatment (neem), untreated maize seed and bare soil were also investigated. Insect health outcomes were also monitored to provide toxicity/mortality data. Wireworms exposed to compound X produced greater burrow networks than untreated maize and neem treatments, similar to that in volume of those produced in bare soil. Compound X exposure also elicited the production of more complex burrow structures, a function of the number of vertices, edges and faces of a shape (V-E+F) related to the number of interconnected branches, compared to any other treatments. Compound X, tefluthrin and thiamethoxam induced mortality at greater rates than neem or untreated, suggesting all three could have potential to manage wireworm populations and reduce yield loss, but only compound X modified burrowing behaviour. With soil biopores playing an important role in soil productivity and carbon sequestration, the wider implications of this increase in burrowing activity for food security and climate change warrants further exploration.

Partial recovery and weak consistency in the non-uniform hypergraph stochastic block model

Abstract We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $\gamma$ fraction of the vertices classified correctly, where $\gamma \in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati ((2017) Ann. Stat.45(1) 289–315.) for non-uniform HSBMs. Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularised adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularisation of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

Switchover phenomenon for general graphs

AbstractWe study SIR‐type epidemics (susceptible‐infected‐resistant) on graphs in two scenarios: (i) when the initial infections start from a well‐connected central region and (ii) when initial infections are distributed uniformly. Previously, Ódor et al. demonstrated on a few random graph models that the expectation of the total number of infections undergoes a switchover phenomenon; the central region is more dangerous for small infection rates, while for large rates, the uniform seeding is expected to infect more nodes. We rigorously prove this claim under mild, deterministic assumptions on the underlying graph. If we further assume that the central region has a large enough expansion, the second moment of the degree distribution is bounded and the number of initial infections is comparable to the number of vertices, the difference between the two scenarios is shown to be macroscopic.

Secondary vertex reconstruction with MaskFormers

In high-energy particle collisions, the reconstruction of secondary vertices from heavy-flavour hadron decays is crucial for identifying and studying jets initiated by b- or c-quarks. Traditional methods, while effective, require extensive manual optimisation and struggle to perform consistently across wide regions of phase space. Meanwhile, recent advancements in machine learning have improved performance but are unable to fully reconstruct multiple vertices. In this work we propose a novel approach to secondary vertex reconstruction based on recent advancements in object detection and computer vision. Our method directly predicts the presence and properties of an arbitrary number of vertices in a single model. This approach overcomes the limitations of existing techniques. Applied to simulated proton-proton collision events, our approach demonstrates significant improvements in vertex finding efficiency, achieving a 10% improvement over an existing state-of-the-art method. Moreover, it enables vertex fitting, providing accurate estimates of key vertex properties such as transverse momentum, radial flight distance, and angular displacement from the jet axis. When integrated into a flavour tagging pipeline, our method yields a 50% improvement in light-jet rejection and a 15% improvement in c-jet rejection at a b-jet selection efficiency of 70%. These results demonstrate the potential of adapting advanced object detection techniques for particle physics, and pave the way for more powerful and flexible reconstruction tools in high-energy physics experiments.

On Broadcast Schemes of Knödel Graphs

Knödel graphs on even number of vertices serve as crucial components in constructing various broadcast graphs, particularly those on an odd number of vertices. The efficacy of these constructions heavily relies on the chosen broadcast scheme. Dimensional broadcast schemes are widely employed for broadcasting Knödel graphs. This paper delves into the investigation of standard and reverse dimensional broadcast schemes identifying numerous valid schemes. Specifically, let [Formula: see text] be a Knödel graph of order [Formula: see text] and [Formula: see text]. We explore the impact of [Formula: see text] cyclic shifts on both standard and reverse dimensional broadcast schemes of Knödel graphs, where [Formula: see text], with repetition of the second dimension.

Polyhedra with Hexagonal and Triangular Faces and Three Faces Around Each Vertex

AbstractWe analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call “trihexes”. Trihexes are analogous to fullerenes, which are 3-regular planar graphs whose faces are all hexagons and pentagons. Every trihex can be represented as the quotient of a hexagonal tiling of the plane under a group of isometries generated by $$180^\circ $$ 180 ∘ rotations. Every trihex can also be described with either one or three “signatures”: triples of numbers that describe the arrangement of the rotocenters of these rotations. Simple arithmetic rules relate the three signatures that describe the same trihex. We obtain a bijection between trihexes and equivalence classes of signatures as defined by these rules. Labeling trihexes with signatures allows us to put bounds on the number of trihexes for a given number of vertices v in terms of the prime factorization of v and to prove a conjecture concerning trihexes that have no “belts” of hexagons.

On Minkowski Addition of Convex Polygons in the Plane

Summary We investigate the Minkowski addition of convex polygons in the plane. It is well-known that the Minkowski addition of two planar convex polygons with m and n vertices is a convex polygon with at most m + n vertices. We give an explicit formula for calculating the exact number of vertices of the resulting polygon.