Adjacent Vertices Research Articles

Local Multiset Dimension of Amalgamation Graphs.

Background: One of the topics of distance in graphs is the resolving set problem. Suppose the set W = { s 1, s 2, …, s k} ⊂ V ( G), the vertex representations of ∈ V ( G) is r m( x| W) = { d( x, s 1), d( x, s 2), …, d( x, s k)}, where d( x, s i) is the length of the shortest path of the vertex x and the vertex in W together with their multiplicity. The set W is called a local m-resolving set of graphs G if r m( v| W)≠ r m( u| W) for uv ∈ E( G). The local m-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of G, denoted by md l( G). Thus, if G has an infinite local multiset dimension and then we write . Methods: This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set W ⊂ V ( G); determine the vertex representation of two adjacent vertices in G; and prove the theorem. Results: The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely md l( Amal( G, v, m)) ≤ m. md l( G) and their exact value of local multiset dimension of some families of graphs namely md l( Amal( P n, v, m)) = 1, , md l( Amal( W n, v, m)) = m. md l( W n), md l( Amal( F n, v, m)) = m. md l( F n) for d( v) = n, . Conclusions: We have found the upper bound of a local multiset dimension. There are some graphs which attain the upper bound of local multiset dimension namely wheel graphs.

A connectivity index based on adjacent vertices in cubic fuzzy graph with an application

A cubic fuzzy graph is a type of fuzzy graph that simultaneously supports two different fuzzy memberships. The study of connectivity in cubic fuzzy graph is an interesting and challenging topic. This research generalized the neighborhood connectivity index in a cubic fuzzy graph with the aim of investigating the connection status of nodes with respect to adjacent vertices. In this survey, the neighborhood connectivity index was introduced in the form of two numerical and distance values. Some characteristics of the neighborhood connectivity index were investigated in cubic fuzzy cycles, saturated cubic fuzzy cycle, complete cubic fuzzy graph and complementary cubic fuzzy graph. The method of constructing a cubic fuzzy graph with arbitrary neighborhood connectivity index was the other point in this research. The results showed that the neighborhood connectivity index depends on the potential of nodes and the number of neighboring nodes. This research was conducted on the Central Bank’s data regarding inter-bank relations and its results were compared in terms of neighborhood connectivity index.

Efficient 2-Distance Coloring Method for Maximum and Average Sparse Graphs in Resource Allocation Optimization of Microgrids

In microgrids management, optimizing resource allocation and enhancing efficiency are critical challenges. This paper introduces the 2-distance coloring technique as a novel approach to address these issues by focusing on the coloring problem of sparse graphs. Through rigorous theoretical derivation, we establish the minimum number of colors required for 2-distance coloring in sparse graph scenarios. Specifically, we demonstrate that for a graph G with an average maximum degree less than 2+17/20 and a maximum degree of 6, it can be list 2-distance colored using no more than 11 colors. This means that, given any set of 11 colors for each vertex, a valid 2-distance coloring can be achieved, ensuring no two adjacent vertices share the same color. This finding is pivotal, as it sets a precedent for the number of resources needed to efficiently manage and allocate resources in microgrids systems through 2-distance coloring. The application of this technique in microgrids promises to revolutionize resource distribution, reducing overlap and maximizing efficiency across the network.

New parallelism and heuristic approaches for generating tree t‐spanners in graphs

SummaryThe ‐admissibility is a min‐max problem that concerns to determine whether a graph contains a spanning tree in which the distance between any two adjacent vertices of is at most in . The stretch index of , , is the smallest for which is ‐admissible. This problem is in P for , NP‐complete for , , and remaining open for . In a very recent development, Couto et al. (Inf Process Lett, 2022; 177: 106265) introduced both sequential and parallel algorithms for constructing spanning trees. Additionally, they proposed two greedy heuristics for generating a candidate solution tree, but they left unresolved the issue of how to decide between two vertices when both have equal chances of being chosen in a greedy step. This criterion is important, since different branches can yield different stretch indexes. In response to this question, we developed nine new heuristics that use the concept of vertex importance in complex networks. Our research evaluates results on several types of graphs, including Barabási‐Albert, Erdős‐Rényi, Watts‐Strogatz, and bipartite graphs. Furthermore, we introduce a new parallel algorithm that employs a method using induced cycle of the graph to compare its performance with previously proposed algorithms. We develop a deep analysis on the proposed strategies (parallel and heuristics) comparing all of them to the other ones in the literature and as a result we obtain the best results so far in order to obtain exact values (or heuristics) of stretch indexes.

On local distance antimagic chromatic number of graphs disjoint union with 1-regular graphs

On local distance antimagic chromatic number of graphs disjoint union with 1-regular graphs

Sombor Index and Sombor Polynomial of the Noncommuting Graph Associated to Some Finite Groups

Sombor index is a newly developed degree-based topological index which involves the degree of the vertex in a simple connected graph. The Sombor index is known as the square root of the sum of the squared degrees of two adjacent vertices in a graph. Meanwhile, the noncommuting graph associated to a group is a graph where its vertices are the non-central elements of the group and two vertices are adjacent if and only if they do not commute. In this study, a new notion called the Sombor polynomial is introduced. Then, the general formula of the Sombor index and the Sombor polynomial of the noncommuting graph associated to some finite groups are determined by using their definitions and some preliminaries. The groups involved in this research are the dihedral groups, the quasidihedral groups, and the generalized quaternion groups. The results found can help the chemists and biologists to predict the chemical and physical properties of the molecules without involving any laboratory work.

NDC Pebbling Number for Some Class of Graphs

Let \(G\) be a connected graph. A pebbling move is defined as taking two pebbles from one vertex and the placing one pebble to an adjacent vertex and throwing away the another pebble. A dominating set \(D\) of a graph \(G=(V,E)\) is a non-split dominating set if the induced graph \(\) is connected. The Non-split Domination Cover(NDC) pebbling number, \(\psi_{ns}(G)\), of a graph $G$ is the minimum of pebbles that must be placed on \(V(G)\) such that after a sequence of pebbling moves, the set of vertices with a pebble forms a non-split dominating set of \(G\), regardless of the initial configuration of pebbles. We discuss some basic results and determine \(\psi_{ns}\) for some families of standard graphs.

A Memetic Algorithm Designed for Solving Graph Coloring Problems: A Round-Robin Sports Scheduling Case Study

The issue of graph coloring is concerned with assigning colors to each vertex of an undirected graph so that adjacent vertices are not assigned the same color. Due to its NP-hard nature, a variety of heuristics and metaheuristics have been developed to tackle this problem. One of these approaches is the memetic algorithm, which was introduced as a solution for this problem. Another example is using a metaheuristic approach to address the round-robin sports scheduling problem. The algorithm suggested for this task comprises three primary components: (1) a randomized danger heuristic algorithm, (2) tabu search, and (3) a genetic algorithm with adaptive multi-parent crossover. To assess the performance of this algorithm, a set of test problems from different benchmark graphs in two categories were selected and compared with nine effective heuristics from existing literature. The results show that the algorithm performs exceptionally well on benchmark graphs that are known to be challenging. Additionally, a case study was conducted to demonstrate the effectiveness of the proposed algorithm

Some New Bounds on the Modified Symmetric Division Deg Index

The use of graph theory in the fields of chemistry, pharmacy, communication, maps, and aeronautics is significant. In order to study the properties of chemical compounds, the molecules of those compounds are modeled as graphs. Boiling point, enthalpy, π -electron energy, and molecular weight are a few examples of physical properties that are related to the geometric structure of the compound. Recently, the modified symmetric division deg (mSDD(G), in short) index is {defined} as the total of all adjacent vertices in pairs μυ of the term 12(dμdν+dνdμ)−−−−−−−−−−−−√. The purpose of this article is to demonstrate the usefulness of mSDD(G) index through the resolution of an interdisciplinary problem describing the structure of benzenoid hydrocarbons. With the help of linear regression models, we have studied the physicochemical properties of benzenoid hydrocarbons. Strong correlations were obtained, and the bounds for the same index were subsequently established.

Graph admissibility: Case generation and analysis by learning models

A graph G is t-admissible if it has a spanning tree T such that any adjacent vertices in G are at a distance at most t in T, in which case T is called a tree t-spanner of G. We denote as the stretch index ofG, or σ(G), the smallest value t such that G is t-admissible. Despite having a polynomial decider for t=2, the problem is NP-complete for t≥4, while the t=3 variant remains open after decades since its proposal.The present work elaborates algorithms to create graphs with known stretch indexes and/or admissibility factors. Indeed, we show how to build a dataset providing the parameter σ of all 12,111 isomorphism classes of connected graphs from 3 to 8 vertices and discuss random generation methods for t-admissible and non-t-admissible graphs of arbitrary sizes. These techniques give us the ability to study the problem from machine learning’s perspective. We show how both lazy and eager models can have high performance under interesting testing conditions, even for instances in the NP-complete spectrum of the problem.

Generation and new infinite families of K2-hypohamiltonian graphs

We present an algorithm which can generate all pairwise non-isomorphic K2-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, of a given order. We introduce new bounding criteria specifically designed for K2-hypohamiltonian graphs, allowing us to improve upon earlier computational results. Specifically, we characterise the orders for which K2-hypohamiltonian graphs exist and improve existing lower bounds on the orders of the smallest planar and the smallest bipartite K2-hypohamiltonian graphs. Furthermore, we describe a new operation for creating K2-hypohamiltonian graphs that preserves planarity under certain conditions and use it to prove the existence of a planar K2-hypohamiltonian graph of order n for every integer n≥134. Additionally, motivated by a theorem of Thomassen on hypohamiltonian graphs, we show the existence K2-hypohamiltonian graphs with large maximum degree and size.

A Solution to Graph Coloring Problem Using Genetic Algorithm

INTRODUCTION: The Graph Coloring Problem (GCP) involves coloring the vertices of a graph in such a way that no two adjacent vertices share the same color while using the minimum number of colors possible.
 OBJECTIVES: The main objective of the study is While keeping the constraint that no two neighbouring vertices have the same colour, the goal is to reduce the number of colours needed to colour a graph's vertices. It further investigate how various techniques impact the execution time as the number of nodes in the graph increases.
 METHODS: In this paper, we propose a novel method of implementing a Genetic Algorithm (GA) to address the GCP.
 RESULTS: When the solution is implemented on a highly specified Google Cloud instance, we likewise see a significant increase in performance. The parallel execution on Google Cloud shows significantly faster execution times than both the serial implementation and the parallel execution on a local workstation. This exemplifies the benefits of cloud computing for computational heavy jobs like GCP.
 CONCLUSION: This study illustrates that a promising solution to the Graph Coloring Problem is provided by Genetic Algorithms. Although the GA-based approach does not provide an optimal result, it frequently produces excellent approximations in a reasonable length of time for a variety of real-world situations.

The Realizability of Theta Graphs as Reconfiguration Graphs of Minimum Independent Dominating Sets

Abstract The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted ℐ (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)-sets are adjacent if and only if they differ by two adjacent vertices. Not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that H = ℐ (G). We consider a class of graphs called “theta graphs”: a theta graph is the union of three internally disjoint nontrivial paths with the same two distinct end vertices. We characterize theta graphs that are i-graph realizable, showing that there are only finitely many that are not. We also characterize those line graphs and claw-free graphs that are i-graphs, and show that all 3-connected cubic bipartite planar graphs are i-graphs.

Inclusion chromatic index of random graphs

AbstractErdős and Wilson proved in 1977 that almost all graphs have chromatic index equal to their maximum degree. In 2001 Balister extended this result and proved that the same number of colours is almost always sufficient if we additionally demand the distinctness of the sets of colours incident with any two vertices. We study a stronger condition and show that one more colour is almost always sufficient and necessary if the inclusion of these sets is forbidden for any pair of adjacent vertices. We also settle the value of a more restrictive graph invariant for almost all graphs, where inclusion is forbidden for all pairs of vertices, which necessitates one more colour for graphs of even order.

On the Spectral Radius and Sombor Energy of the Non-Commuting Graph for Dihedral Groups

The non-commuting graph, denoted by , is defined on a finite group , with its vertices are elements of excluding those in the center of . In this graph, two distinct vertices are adjacent whenever they do not commute in . The graph can be associated with several matrices including the most basic matrix, which is the adjacency matrix, , and a matrix called Sombor matrix, denoted by . The entries of are either the square root of the sum of the squares of degrees of two distinct adjacent vertices, or zero otherwise. Consequently, the adjacency and Sombor energies of is the sum of the absolute eigenvalues of the adjacency and Sombor matrices of , respectively, whereas the spectral radius of is the maximum absolute eigenvalues. Throughout this paper, we find the spectral radius obtained from the spectrum of and the Sombor energy of for dihedral groups of order , where . Moreover, there is an almost linear correlation between the Sombor energy and the adjacency energy of for which is slightly different than reported earlier in previous literature.

Total Face Irregularity Strength of Certain Graphs

The edge k-labeling ψ of G is defined by a mapping from EG to a set of integers 1,2,…,k, where the integer weight assigned to the vertex x∈VG is given as wψx=∑ψxy, such that the sum is taken over every vertex of y∈VG that is adjacent to x and the integer weights of adjacent vertices must be distinct for all vertices with x≠y. An irregular assignment of G using atmost k labels which is considered to be a minimum k is defined as irregularity strength of a graph G and can be denoted as sG. There are also further works on familiar irregular assignments, such as edge irregular labelings, vertex irregular total labelings, edge irregular total labelings, and face irregular entire k-labelings of plane graphs. A plane graph can be defined as a graph that is embedded in the plane in which no two lines will be intersected. In a plane graph the number of regions present are called faces and we denote it as F. The concept of total face irregularity strength is defined by the motivation of irregular networks and entire irregular face k-labeling. In our paper, we have obtained a minimum bound for the total face irregularity strength of two-connected plane graphs like cycle-of-ladder, C-necklace graph, P-necklace graph, sibling tree, and triangular graph.

Generalized Splines on Graphs with Two Labels and Polynomial Splines on Cycles

Generalized splines are an algebraic combinatorial framework that generalizes and unifies various established concepts across different fields, most notably the classical notion of splines and the topological notion of GKM theory. The former consists of piecewise polynomials on a combinatorial geometric object like a polytope, whose polynomial pieces agree to a specified degree of differentiability. The latter is a graph-theoretic construction of torus-equivariant cohomology that Shareshian and Wachs used to reformulate the well-known Stanley-Stembridge conjecture, a reformulation that was recently proven to hold by Brosnan and Chow and independently Guay-Paquet.
 This paper focuses on the theory of generalized splines. A generalized spline on a graph $G$ with each edge labeled by an ideal in a ring $R$ consists of a vertex-labeling by elements of $R$ so that the labels on adjacent vertices $u, v$ differ by an element of the ideal associated to the edge $uv$. We study the $R$-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: $1)$ for all graphs when the set of possible edge-labelings consists of at most two finitely-generated ideals, and $2)$ for cycles when the set of possible edge-labelings consists of principal ideals generated by elements of the form $(ax+by)^2$ in the polynomial ring $\mathbb{C}[x,y]$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in the theory of classical (analytic) splines.

2-tone coloring of cactus graphs

A 2-tone coloring of a graph assigns two distinct colors to each vertex with the restriction that adjacent vertices have no common colors, and vertices at distance two have at most one common color. The 2-tone chromatic number of a graph is the minimum number of colors in any 2-tone coloring. A cactus graph has every block a cycle or edge. We determine the 2-tone chromatic number of all cactus graphs with maximum degree Δ≠6. When Δ=6, we determine the 2-tone chromatic number of all cactus graphs that are triangle-free and those with circumference at most 8.

Geometric approach to vertex‐degree‐based topological indices–Elliptic Sombor index, theory and application

AbstractA novel geometric method is proposed for constructing vertex‐degree‐based molecular structure descriptors (topological indices). The model is based on an ellipse whose focal points represent the degrees of a pair of adjacent vertices. This approach enables a geometric interpretation of several previously known topological indices, and lead to design of a few new. The area of the ellipse induces a vertex‐degree‐based topological index of remarkable simplicity, which we call elliptic Sombor index(). The main mathematical properties of are established, especially its relations to other, earlier known, indices. Then the applicative potential of is analyzed.

On the spectra and spectral radii of token graphs

Let G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by 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}, has as vertices the nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${n\\atopwithdelims ()k}$$\\end{document}k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, Fk(Kn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(K_n)$$\\end{document} is the Johnson graph J(n, k), which is a distance-regular graph used in coding theory. In this paper, we present some results concerning the (adjacency and Laplacian) spectrum 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} in terms of the spectrum of G. For instance, when G is walk-regular, an exact value for the spectral radius ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} (or maximum eigenvalue) 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} is obtained. When G is distance-regular, other eigenvalues of its 2-token graph are derived using the theory of equitable partitions. A generalization of Aldous’ spectral gap conjecture (which is now a theorem) is proposed.