Number Of Vertices Research Articles

Connection-Based Zagreb Indices of Polycyclic Aromatic Hydrocarbons Structures.

Topological indices (TIs) are mathematical formulas that are applied in mathematical chemistry to predict the physical and chemical properties of various chemical structures. In this study, three different types of polycyclic aromatic hydrocarbon structures (PAHs) (i.e., Hexa-peri-hexabenzocoronene, Dodeca-benzo-circumcoronene, and Hexa-cata- hexabenzocoronene) are studied with the help of the different connection number-based Zagreb indices. ϑ = (V(ϑ),E(ϑ)) is used as a graph, where V(ϑ) is a collection of vertices and E(ϑ) is a collection of edges. For a vertex y, ∈V(ϑ), the degree d_ϑ (y), is the number of those vertices that are at a distance of 1 from y and the connection number ρ_ϑ (y) is the number of such vertices that are at a distance of 2 from vertex y. Theoretical applications of topological indices were described in detail. Finally, we obtained the first and second Zagreb connections as well as the modified first, second, third, and fourth Zagreb connection indices, which were calculated for three different types (Hexa-peri-hexabenzocorone, Dodeca-benzo-circumcoronene, and Hexa-cata-hexabenzocoronene) of polycyclic aromatic hydrocarbon structures.

Vertex-primitive digraphs with large fixity

AbstractThe relative fixity of a digraph $$\Gamma $$ Γ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of $$\Gamma $$ Γ and the number of vertices of $$\Gamma $$ Γ . We characterize the vertex-primitive digraphs whose relative fixity is at least $$\frac{1}{3}$$ 1 3 , and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant.

Classification of schizophrenic traits in transcriptions of audio spectra from patient literature: artificial intelligence models enhanced by geometric properties

Schizophrenia is a severe mental illness that affects approximately 1% of the global population and presents significant challenges for patients, families, and healthcare professionals. Characterized by symptoms such as delusions, hallucinations, disorganized speech or behavior, and cognitive impairment, this condition has an early onset and chronic trajectory, making it a debilitating challenge. Schizophrenia also imposes a substantial burden on society, exacerbated by the stigma associated with mental disorders. Technological advancements, such as computerized semantic, linguistic, and acoustic analyses, are revolutionizing the understanding and assessment of communication alterations, a significant aspect in various severe mental illnesses. Early and accurate diagnosis is crucial for improving prognosis and implementing appropriate treatments. In this context, the advancement of Artificial Intelligence (AI) has provided new perspectives for the treatment of schizophrenia, with machine learning techniques and natural language processing allowing a more detailed analysis of clinical, neurological, and behavioral data sets. The present article aims to present a proposal for computational models for the identification of schizophrenic traits in texts.  The database used in this article was created with 139 excerpts of patients' speeches reported in the book “Memories of My Nervous Disease” by German judge Daniel Paul Schreber, classifying them into three categories: 1 - schizophrenic, 2 - with schizophrenic traits and 3 - without any relation to the disorder. Of these speeches, 104 were used for training the models and the others 35 for validation.Three classification models were implemented using features based on geometric properties of graphs (number of vertices, number of cycles, girth, vertex of maximum degree, maximum clique size) and text entropy. Promising results were observed in the classification, with the Decision Tree-based model [1] achieving 100% accuracy, the KNN- k-Nearest Neighbor model observed with 62.8% accuracy, and the 'centrality-based' model with 59% precision. The high precision rates, observed when geometric properties are incorporated into Artificial Intelligence Models, suggest that the models can be improved to the point of capturing the language deviation traits that are indicative of schizophrenic disorders. In summary, this study paves the way for significant advances in the use of geometric properties in the field of psychiatry, offering a new data-based approach to the understanding and therapy of schizophrenia.

Unified approach to the inverse median location problem under the sum and the max objectives applying ordered median function

ABSTRACT In this paper, we propose a unified approach to solving the inverse 1-median problem on trees under both the sum and max objectives by using the ordered median function. For the problem under the k-centrum cost function, we investigate the structure of an optimal solution and transform the problem into a univariate optimization problem where the parameter equals the k largest cost. We take advantage of the convexity of the objective function to efficiently find the optimal solution in O ( nlog ⁡ n ) time, where n is the number of vertices in the tree. For the problem under the centdian cost function, we parameterize it based on the maximum cost item and prove that the induced cost function is convex. We leverage this insight to develop a combinatorial algorithm that solves the corresponding problem in O ( nlog ⁡ n ) time.

Hub Parameters of Hypergraph

A hypergraph is an extension of a graph in which one edge can connect any number of vertices. In contrary, an edge connects exactly two vertices in a graph.In this paper we introduce hub-hyperpath, hubset and hubnumber of a hypergraph. Also we defined the hubnumber of a different types of hypergraphs and analyze some of its properties. Then the hub number of a hypergraph is compared with its dual hypergraph. Hubset can be useful for various tasks, such as targeted marketing and social networks. Additionally, finding the hub number of a graph is useful in network security, as it helps in identifying nodes that, if compromised, could have a significant impact on the overall network.

On a Simple Connection Between $$\Delta$$-Modular ILP and LP, and a New Bound on the Number of Integer Vertices

On a Simple Connection Between $$\Delta$$-Modular ILP and LP, and a New Bound on the Number of Integer Vertices

Chromatic numbers of certain hypercube variants

The total (vertex/edge) chromatic number of a graph G, χ ″ ( G ) ( χ ( G ) / χ ′ ( G ) ) is defined to be the minimum number of colours needed to colour the vertices and edges, together termed as elements (vertices/edges) of a graph in such a way that no two adjacent/incident elements (vertices/edges) are given the same colour. In this article, we determine the vertex, edge and total chromatic number of some hypercube variants such as folded hypercube, enhanced hypercube, augmented hypercube and exchanged hypercube.

Graphs with minimum degree-entropy

We continue studying extremal values of the degree-entropy, which is an information-theoretic measure defined as the Shannon entropy based on the information functional involving vertex degrees. For a graph with a given number of vertices and edges achieving the minimum entropy value, we show its unique structure. Also, a tight lower bound for the entropy in bipartite graphs with a given number of vertices and edges is proved. Our result directly derive the result of Cao et al. (2014) that for a tree with a given number of vertices, the minimum value of the entropy is attained if and only if the tree is the star.

Eigenvalue multiplicity of a graph in terms of the number of external vertices

The multiplicity of an eigenvalue λ of a graph G is denoted by m(G,λ). In a connected graph G with at least two vertices, a vertex is called external if it is not a cut vertex. In a tree, an external vertex is exactly a pendant vertex. Let ϵ(G) be the number of external vertices in G. In this paper, we prove that m(G,λ)≤ϵ(G)−1 for any λ and characterize the extremal graphs with m(G,−1)=ϵ(G)−1, which generalizes the main result of Wang et al. [Linear Multi. Alg., 2020] from a tree to an arbitrary connected graph.

Extremal results on the Mostar index of trees with fixed parameters

For a graph [Formula: see text], the Mostar index of [Formula: see text] is the sum of [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the number of vertices of [Formula: see text] that have a smaller distance in [Formula: see text] to [Formula: see text] than to [Formula: see text], and analogously for [Formula: see text]. We determine all the graphs that maximize and minimize the Mostar index, respectively, over all trees in terms of some fixed parameters like the number of odd vertices, the number of vertices of degree two, and the number of pendent paths of fixed length.

Equitable colorings of ��-corona products of cubic graphs

A graph G is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest integer k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by x=( G). In this paper the problem of determining the value of equitable chromatic number for multicoronas of cubic graphs G◦ lH is studied. The problem of ordinary coloring of multicoronas of cubic graphs is solvable in polynomial time. The complexity of equitable coloring problem is an open question for these graphs. We provide some polynomially solvable cases of cubical multicoronas and give simple linear time algorithms for equitable coloring of such graphs which use at most x=( G◦ lH) + 1 colors in the remaining cases.

Theoretical Aspects of Generating Instances with Unique Solutions: Pre-assignment Models for Unique Vertex Cover

The uniqueness of an optimal solution to a combinatorial optimization problem attracts many fields of researchers' attention because it has a wide range of applications, it is related to important classes in computational complexity, and the existence of only one solution is often critical for algorithm designs in theory. However, as the authors know, there is no major benchmark set consisting of only instances with unique solutions, and no algorithm generating instances with unique solutions is known; a systematic approach to getting a problem instance guaranteed having a unique solution would be helpful. A possible approach is as follows: Given a problem instance, we specify a small part of a solution in advance so that only one optimal solution meets the specification. This paper formulates such a ``pre-assignment'' approach for the vertex cover problem as a typical combinatorial optimization problem and discusses its computational complexity. First, we show that the problem is ΣP2-complete in general, while the problem becomes NP-complete when an input graph is bipartite. We then present an O(2.1996^n)-time algorithm for general graphs and an O(1.9181^n)-time algorithm for bipartite graphs, where n is the number of vertices. The latter is based on an FPT algorithm with O*(3.6791^τ) time for vertex cover number τ. Furthermore, we show that the problem for trees can be solved in O(1.4143^n) time.

On the complexity of nucleolus computation for bipartite b-matching games

We explore the complexity of nucleolus computation in b-matching games on bipartite graphs. We show that computing the nucleolus of a simple b-matching game is NP-hard when b≡3 even on bipartite graphs of maximum degree 7. We complement this with partial positive results in the special case where b values are bounded by 2. In particular, we describe an efficient algorithm when a constant number of vertices satisfy bv=2 as well as an efficient algorithm for computing the non-simple b-matching nucleolus when b≡2.

Pass-Efficient Algorithms for Graph Spectral Clustering (Student Abstract)

Graph spectral clustering is a fundamental technique in data analysis, which utilizes eigenpairs of the Laplacian matrix to partition graph vertices into clusters. However, classical spectral clustering algorithms require eigendecomposition of the Laplacian matrix, which has cubic time complexity. In this work, we describe pass-efficient spectral clustering algorithms that leverage recent advances in randomized eigendecomposition and the structure of the graph vertex-edge matrix. Furthermore, we derive formulas for their efficient implementation. The resulting algorithms have a linear time complexity with respect to the number of vertices and edges and pass over the graph constant times, making them suitable for processing large graphs stored on slow memory. Experiments validate the accuracy and efficiency of the algorithms.

Generalized Approach to Optimal Polylinearization for Smart Sensors and Internet of Things Devices

This study introduces an innovative numerical approach for polylinear approximation (polylinearization) of non-self-intersecting compact sensor characteristics (transfer functions) specified either pointwise or analytically. The goal is to partition the sensor characteristic optimally, i.e., to select the vertices of the approximating polyline (approximant) along with their positions, on the sensor characteristics so that the distance (i.e., the separation) between the approximant and the characteristic is rendered below a certain problem-specific tolerance. To achieve this goal, two alternative nonlinear optimization problems are solved, which differ in the adopted quantitative measure of the separation between the transfer function and the approximant. In the first problem, which relates to absolutely integrable sensor characteristics (their energy is not necessarily finite, but they can be represented in terms of convergent Fourier series), the polylinearization is constructed by the numerical minimization of the L1-metric (a distance-based separation measure), concerning the number of polyline vertices and their locations. In the second problem, which covers the quadratically integrable sensor characteristics (whose energy is finite, but they do not necessarily admit a representation in terms of convergent Fourier series), the polylinearization is constructed by numerically minimizing the L2-metric (area- or energy-based separation measure) for the same set of optimization variables—the locations and the number of polyline vertices.

A characterization of star-perfect graphs

Motivated by Berge perfect graphs, we define star-perfect graphs and characterize them. For a finite simple graph G(V, E), let θ s ( G ) denote the minimum number of induced stars contained in G such that the union of their vertex sets is V(G), and let α s ( G ) denote the maximum number of vertices in G such that no two of them are contained in the same induced star of G. We call a graph G star-perfect if α s ( H ) = θ s ( H ) , for every induced subgraph H of G. A graph G is star-perfect if and only if G is ( C 3 , C 3 k + 1 , C 3 k + 2 ) -free, for every k ≥ 1 . A bipartite graph G is star-perfect if and only if every induced cycle in G is of length 6 k , k ≥ 1 . The minimum parameter θ s ( G ) and the maximum parameter α s ( G ) have been extensively studied in various contexts.

The hardness of recognising poorly matchable graphs and the hunting of the d-snark

Abstract. An r-graph is an r-regular graph G on an even number of vertices where every odd set X ⊆V(G) is connected by at least r edges to its complement V(G) \ X. Every r-graph has a perfect matching and in a poorly matchable r-graph every pair of perfect matchings intersect, which implies that poorly matchable r-graphs are not r-edge-colourable. We prove, for each fixed r ≥ 3, that poorly matchable r-graph recognition is coNP-complete, an indication that, for each odd d ≥ 3, it may be a hard problem to recognise d-regular (d−1)-edge-connected non-d-edge-colourable graphs, referred to as d-snarks in this paper. We show how to construct, for every fixed odd d ≥ 5, an infinite family of d-snarks. These families provide a natural extension to the well-known Loupekine snarks. We also discuss how the hunting of the smallest d-snarks may help in strengthening and better understanding the major Overfull Conjecture on edge-colouring simple graphs.

Flip distance and triangulations of a polyhedron

AbstractIt is known that the flip distance between two triangulations of a convex polygon is related to the smallest number of tetrahedra in the triangulation of some polyhedron. The latter was used to compute the diameter of the flip graph of convex polygons with a large number of vertices. However, it is yet unknown whether the flip distance and this smallest number of tetrahedra are always the same or even close. In this work, we find examples to show that the ratio between these two numbers can be arbitrarily close to . We also propose two conjectures in the end, one about this ratio, and the other may have some implications on when two triangulations can achieve maximal distance.

FulBM: Fast fully batch maintenance for landmark-based 3-hop cover labeling

Landmark-based 3-hop cover labeling is a category of approaches for shortest distance/path queries on large-scale complex networks. It pre-computes an index offline to accelerate the online distance/path query. Most real-world graphs undergo rapid changes in topology, which makes index maintenance on dynamic graphs necessary. So far, the majority of index maintenance methods can handle only one edge update (either an addition or deletion) each time. To keep up with frequently changing graphs, we research the ful ly b atch m aintenance problem for the 3-hop cover labeling, and proposed the method called FulBM . FulBM is composed of two algorithms: InsBM and DelBM, which are designed to handle batch edge insertions and deletions respectively. This separation is motivated by the insight that batch maintenance for edge insertions are much more time-efficient, and the fact that most edge updates in the real world are incremental. Both InsBM and DelBM are equipped with well-designed pruning strategies to minimize the number of vertex accesses. We have conducted comprehensive experiments on both synthetic and real-world graphs to verify the efficiency of FulBM and its variants for weighted graphs. The results show that our methods achieve 5.5 × to 228 × speedup compared with the state-of-the-art method.

Equilibria in schelling games: computational hardness and robustness

In the simplest game-theoretic formulation of Schelling’s model of segregation on graphs, agents of two different types each select their own vertex in a given graph so as to maximize the fraction of agents of their type in their occupied neighborhood. Two ways of modeling agent movement here are either to allow two agents to swap their vertices or to allow an agent to jump to a free vertex. The contributions of this paper are twofold. First, we prove that deciding the existence of a swap-equilibrium and a jump-equilibrium in this simplest model of Schelling games is NP-hard, thereby answering questions left open by Agarwal et al. [AAAI ’20] and Elkind et al. [IJCAI ’19]. Second, we introduce two measures for the robustness of equilibria in Schelling games in terms of the minimum number of edges or the minimum number of vertices that need to be deleted to make an equilibrium unstable. We prove tight lower and upper bounds on the edge- and vertex-robustness of swap-equilibria in Schelling games on different graph classes.