Adjacent Vertices Research Articles

Prime Labeling of Union of Some Graphs

A prime labeling of a graph G is a map from the vertex set of G to the set {1, 2, ..., |V (G)|} such that any two adjacent vertices in the graph G have labels that are relatively prime. In this paper, we discuss when the disjoint union of some graphs is a prime graph.

Sequence of Bounds for Spectral Radius and Energy of Digraph

The graph spectra analyze the structure of the graph using eigenspectra. The spectral graph theory deals with the investigation of graphs in terms of the eigenspectrum. In this paper, the sequence of lower bounds for the spectral radius of digraph D having at least one doubly adjacent vertex in terms of indegree is proposed. Particularly, it is exhibited that ρ(D)≥αj=∑p=1m(χj+1(p))2∑p=1m(χj(p))2, such that equality is attained iff D=G↔+ {DE∉ Cycle}, where each component of associated graph is a k-regular or (k1,k2) semiregular bipartite. By utilizing the sequence of lower bounds of the spectral radius of D, the sequence of upper bounds of energy of D, where the sequence decreases when eU≤αj and increases when eU>αj, are also proposed. All of the obtained inequalities are elaborated using examples. We also discuss the monotonicity of these sequences.

Optimal L(d,1)-Labeling of Certain Direct Graph Bundles Cycles over Cycles and Cartesian Graph Bundles Cycles over Cycles

An L(d,1)-labeling of a graph G=(V,E) is a function f from the vertex set V(G) to the set of nonnegative integers such that the labels on adjacent vertices differ by at least d and the labels on vertices at distance two differ by at least one, where d≥1. The span of f is the difference between the largest and the smallest numbers in f(V). The λ1d-number of G, denoted by λ1d(G), is the minimum span over all L(d,1)-labelings of G. We prove that λ1d(X)≤2d+2, with equality if 1≤d≤4, for direct graph bundle X=Cm×σℓCn and Cartesian graph bundle X=Cm□σℓCn, if certain conditions are imposed on the lengths of the cycles and on the cyclic ℓ-shift σℓ.

Hub Cover Pebbling Number

The hub cover pebbling number, h ∗ ( G ) , of a graph $G$, is the least non-negative integer such that from all distributions of h ∗ ( G ) pebbles over the vertices of G , it is possible to place at least one pebble each on every vertex of a set of vertices of a hub set for G using a sequence of pebbling move operations, each pebbling move operation removes two pebbles from a vertex and places one pebble on an adjacent vertex. Here we compute the hub cover pebbling number for wheel related graphs.

The Multiplicative Sum Zagreb Indices of Graphs with Given Clique Number

The multiplicative sum Zagreb index is a modified version of the well-known Zagreb indices. The multiplicative sum Zagreb index of a graph G is the product of the sums of the degrees of pairs of adjacent vertices. The mathematical properties of the multiplicative sum Zagreb index of graphs with given graph parameters deserve further study, as they can be used to detect chemical compounds and study network structures in mathematical chemistry. Therefore, in this paper, the maximal and minimal values of the multiplicative sum Zagreb indices of graphs with a given clique number are presented. Furthermore, the corresponding extremal graphs are characterized.

Adjacent vertex distinguishing total coloring of corona product of graphs

Adjacent vertex distinguishing total coloring of corona product of graphs

Complete characterization of bridge graphs with local antimagic chromatic number 2

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f\colon E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we characterize $s$-bridge graphs with local antimagic chromatic number 2.

Finding the minimum [formula omitted]-weighted dominating sets using heuristic algorithms

In this work, we propose, analyze, and solve a generalization of the k-dominating set problem in a graph, when we consider a weighted graph. Given a graph with weights in its edges, a set of vertices is a k-weighted dominating set if for every vertex outside the set, the sum of the weights from it to its adjacent vertices in the set is bigger than or equal to k. The k-weighted domination number is the minimum cardinality among all k-weighted dominating sets. Since the problem of finding the k-weighted domination number is NP-hard, we have proposed several problem-adapted construction and reconstruction techniques and embedded them in an Iterated Greedy algorithm. The resulting sixteen variants of the Iterated Greedy algorithm have been compared with an exact algorithm. Computational results show that the proposal is able to find optimal or near-optimal solutions within a short computational time. To the best of our knowledge, the k-weighted dominating set problem has never been studied before in the literature and, therefore, there is no other state-of-the-art algorithm to solve it. We have also included a comparison with a particular case of our problem, the minimum dominating set problem and, on average, we achieve same quality results within around 50% of computation time.

HYPERTILING — a high performance Python library for the generation and visualization of hyperbolic lattices

HYPERTILING is a high-performance Python library for the generation and visualization of regular hyperbolic lattices embedded in the Poincaré disk model. Using highly optimized, efficient algorithms, hyperbolic tilings with millions of vertices can be created in a matter of minutes on a single workstation computer. Facilities including computation of adjacent vertices, dynamic lattice manipulation, refinements, as well as powerful plotting and animation capabilities are provided to support advanced uses of hyperbolic graphs. In this manuscript, we present a comprehensive exploration of the package, encompassing its mathematical foundations, usage examples, applications, and a detailed description of its implementation.

Codebase release 1.3 for HYPERTILING

HYPERTILING is a high-performance Python library for the generation and visualization of regular hyperbolic lattices embedded in the Poincaré disk model. Using highly optimized, efficient algorithms, hyperbolic tilings with millions of vertices can be created in a matter of minutes on a single workstation computer. Facilities including computation of adjacent vertices, dynamic lattice manipulation, refinements, as well as powerful plotting and animation capabilities are provided to support advanced uses of hyperbolic graphs. In this manuscript, we present a comprehensive exploration of the package, encompassing its mathematical foundations, usage examples, applications, and a detailed description of its implementation.

Proteinaceous Nanoshells with Quasicrystalline Local Order

Among various proteinaceous nanocontainers and nanoparticles, the most promising ones for various applications in nano- and medical science appear to be those whose structures differ fundamentally from icosahedral viral capsids described by the paradigmatic Caspar-Klug model. By analyzing such anomalous assemblies represented in the Protein Data Bank, we identify a series of shells with square-triangular local order and find that most of them originate from short-period approximants of a dodecagonal tiling consisting of square and triangular tiles. Examining the nonequilibrium assembly of such packings, we propose a new method for obtaining periodic square-triangle approximants and then construct the simplest models of tetragonal, octahedral, and icosahedral shells based on cubic and icosahedral nets cut from the approximant structures. Since gluing the nets can change the distances between adjacent vertices of the resulting shell, we introduce an effective energy, the minimization of which equalizes these distances. While the obtained spherical polyhedra reproduce the structures of experimentally observed protein shells and nanoparticles, the principles of protein organization that we lay out, and the ensuing structural models, can help to discover and investigate similar systems in the future. Published by the American Physical Society 2024

Atomically precise inorganic helices with a programmable irrational twist.

Helicity in solids often arises from the precise ordering of cooperative intra- and intermolecular interactions unique to natural, organic or molecular systems. This exclusivity limited the realization of helicity and its ensuing properties in dense inorganic solids. Here we report that Ga atoms in GaSeI, a representative III-VI-VII one-dimensional (1D) van der Waals crystal, manifest the rare Boerdijk-Coxeter helix motif. This motif is a non-repeating geometric pattern characterized by 1D face-sharing tetrahedra whose adjacent vertices are rotated by an irrational angle. Using InSeI and GaSeI, we show that the modularity of 1D van der Waals lattices accommodates the systematic twisting of a periodic tetrahelix with a 41 screw axis in InSeI to an infinitely extending Boerdijk-Coxeter helix in GaSeI. GaSeI crystals are non-centrosymmetric, optically active and exfoliable to a single chain. These results present a materials platform towards understanding the origin and physical manifestation of aperiodic helicity in low-dimensional solids.

The number of 1-nearly independent vertex subsets

Let G be a graph with vertex set V(G) and edge set E(G). A subset I of V(G) is an independent vertex subset if no two vertices in I are adjacent in G. We study the number, σ 1(G), of all subsets of V(G) that contain exactly one pair of adjacent vertices. We call those subsets 1-nearly independent vertex subsets. Recursive formulas of σ 1 are provided, as well as some cases of explicit formulas. We prove a tight lower (resp. upper) bound on σ 1 for graphs of order n. We deduce as a corollary that the star K 1,n−1 (the tree with degree sequence (n − 1, 1, . , 1)) is the n-vertex tree with smallest σ 1, while it is well known that K1,n−1 is the n-vertex tree with largest number of independent subsets.

Restricted optimal pebbling is NP-hard

Consider a distribution of pebbles on a graph. A pebbling move removes two pebbles from a vertex and place one at an adjacent vertex. A vertex is reachable under a pebble distribution if it has a pebble after the application of a sequence of pebbling moves. A pebble distribution is solvable if each vertex is reachable under it. The size of a pebble distribution is the total number of pebbles. The optimal pebbling number π∗(G) is the size of the smallest solvable distribution. A t-restricted pebble distribution places at most t pebbles at each vertex. The t-restricted optimal pebbling number πt∗(G) is the size of the smallest solvable t-restricted pebble distribution. We show that deciding whether π2∗(G)≤k is NP-complete. We prove that πt∗(G)=π∗(G) if δ(G)≥2|V(G)|3−1 and we show infinitely many graphs which satisfies δ(H)≈12|V(H)| but πt∗(H)≠π∗(H), where δ denotes the minimum degree.

The Recognition of Aerobics Movements Using Bone Data Combined with ST-GCN

To solve the aerobics action recognition and promote the gradual intelligence and standardization of aerobics teaching and evaluation, a network model based on spatial temporal graph convolution and combined with attention mechanism was proposed. This model improved the extraction efficiency of spatiotemporal features and channel features by introducing spatiotemporal graph and channel attention mechanisms, respectively, thereby improving the accuracy of action recognition. And a time extension module was introduced into each basic module, and additional features between adjacent vertices were extracted by extending the time graph between frames. These experiments confirmed that this model exhibited high accuracy in identifying aerobics movements. The recognition accuracy of basic actions was above 93.6%, and the recognition accuracy of two actions had reached 98.4% and 98.6% respectively. For advanced actions, the recognition accuracy of the model had slightly decreased, but the average value was still above 95%. The accuracy of difficult motion recognition had also achieved good results, reaching a maximum of 94.5%. These data indicate that this model can achieve high accuracy in handling action recognition tasks of different difficulty levels, and can identify aerobics movements of different difficulty levels.

On the complexity of distance-d independent set reconfiguration

For a fixed positive integer d≥2, a distance-d independent set (DdIS) of a graph is a vertex-subset whose distance between any two members is at least d. Imagine that there is a token placed on each member of a DdIS. Two DdISs are adjacent under Token Sliding (TS) if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping (TJ), the target vertex needs not to be adjacent to the original one. The Distance-dIndependent Set Reconfiguration (DdISR) problem under TS/TJ asks if there is a corresponding sequence of adjacent DdISs that transforms one given DdIS into another. The problem for d=2, also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of DdISR on different graphs under TS and TJ for any fixed d≥3. On chordal graphs, we show that DdISR under TJ is in P when d is even and PSPACE-complete when d is odd. On split graphs, there is an interesting complexity dichotomy: DdISR is PSPACE-complete for d=2 but in P for d=3 under TS, while under TJ it is in P for d=2 but PSPACE-complete for d=3. Additionally, certain well-known hardness results for d=2 on perfect graphs and planar graphs of maximum degree three and bounded bandwidth can be extended for d≥3.

Computing Topological Indices of Certain Networks

In mathematical chemistry, topological indices are molecular descriptors that are calculated on the molecular graph of a chemical compound. The molecular graph is a graph which is obtained from some chemical structures. The degree of every molecular graph cannot exceeds 4. Topological indices are numerical quantities of a graph that describe its topology. An atom represents a vertex and a bond between two atoms represents an edge in a molecular graph. Mainly there are three types of topological indices viz., degree-based, distance based and eigenvalue-based topological indices. The first degree-based topological indices are the first and second Zagreb indices. The first Zagreb index M_1 is defined as the sum of squares of degrees of each vertex in a graph G and the second Zagreb index M_2 is the product of degree of every adjacent vertices. In this case the summation goes on the set of edges of a graph G. The most studied topological indices are degree-based topological indices. Motivated by these topological indices in this paper, we introduce five new degree-based topological indices based on the neighborhood degree of a vertex. Further, we compute the values of various nanostructures like hexagonal parallelogram P(m,n) nanotube, triangular benzenoid G_n,zigzag-edge coronoid fused with starphene nanotubes ZCS(k,l,m), dominating derived networks D_1,D_2,D_3, Porphyrin Dendrimer, Zinc-Porphyrin Dendrimer, Propyl Ether Imine Dendrimer, Poly(Ethylene amido amine Dendrimer, PAMAM dendrimers(????????1,????????2,????????1), linear polyomino chain L_n,Z_n,B_n^1 (n≥3),B_n^2 (n≥3) and triangular, hourglass, and jagged-rectangle benzenoid systems of these indices. The standard computational techniques are used for the computation of topological indices of nanostructures. For the edge partition of the nanostructures the algebraic techniques are used. Using these techniques computation of topological indices became easy and also helped to get the more accurate results.

On Edge Prime Index of a Graph

Relatively Prime Edge labeling extends the notion of prime labeling by considering edges. Prime labeling requires adjacent vertices to possess relatively prime labels, while relatively prime edge labeling requires adjacent edges to have relatively prime labels. The transformation of a coprime edge-labeled graph into a relatively prime edge-labeled graph introduces the concept of Edge Prime Index (or Relatively Prime Index). This study focuses on cases where a coprime edge-labeled graph can be converted into a relatively prime edge-labeled graph by removing certain edges from graph G, thereby establishing the concept of Edge Prime Index. Finally, the Edge Prime Index of some graphs are found.

On the algebraic connectivity of some token graphs

The k-token graph Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It was proved that the algebraic connectivity of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} equals the algebraic connectivity of G with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} equals the one of G for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph.

The Euler Sombor index of a graph

AbstractA novel geometric method is proposed for constructing vertex‐degree‐based molecular structure descriptors, which is the perimeter of an ellipse whose focal points represent the degree‐point and its dual‐point of a pair of adjacent vertices in a graph . We first analyze the chemical applicability of this elliptic Sombor index, and then establish its mathematical properties and determine the extremal values for this elliptic Sombor index among all (molecular) trees and characterize the corresponding extremal graphs.