Pair Of Vertices Research Articles

One-to-one Disjoint-path Covers of Leaf-sort Graphs

Parallel paths of interconnection networks have attracted much attention in parallel computing systems, they are studied in terms of disjoint paths in an undirected graph [Formula: see text]. A [Formula: see text]-disjoint path cover (k-DPC for short) of a graph [Formula: see text] is a set of [Formula: see text] (internally) disjoint paths that altogether cover every vertice of the graph. A bipartite graph [Formula: see text] is one-to-one bi-k-DPC if there is a k-DPC between any pair of vertices in different partite sets. A bipartite graph is hamiltonian laceable if there exists a hamiltonian path between any pair of vertices in different partite sets. The [Formula: see text]-dimensional leaf-sort graph [Formula: see text] has many good properties, including bipartite, symmetry, vertex-transitive. In this paper, we prove that [Formula: see text] has one-to-one bi-[Formula: see text]-DPC between any two vertices [Formula: see text] and [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] in different partite sets, and [Formula: see text] is hamiltonian laceable.

Variety of General Position Problems in Graphs

Let X be a vertex subset of a graph G. Then u,v∈V(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u, v\\in V(G)$$\\end{document} are X-positionable if V(P)∩X⊆{u,v}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(P)\\cap X \\subseteq \\{u,v\\}$$\\end{document} holds for any shortest u, v-path P. If each two vertices from X are X-positionable, then X is a general position set. The general position number of G is the cardinality of a largest general position set of G and has been already well investigated. In this paper a variety of general position problems is introduced based on which natural pairs of vertices are required to be X-positionable. This yields the total (resp. dual, outer) general position number. It is proved that the total general position sets coincide with sets of simplicial vertices, and that the outer general position sets coincide with sets of mutually maximally distant vertices. It is shown that a general position set is a dual general position set if and only if its complement is convex. Several sufficient conditions are presented that guarantee that a given graph has no dual general position set. The total general position number, the outer general position number, and the dual general position number of arbitrary Cartesian products are determined.

Edge‐arc‐disjoint paths in semicomplete mixed graphs

AbstractThe so‐called weak‐2‐linkage problem asks for a given digraph and distinct vertices of whether has arc‐disjoint paths so that is an ‐path for . This problem is NP‐complete for general digraphs but the first author showed that the problem is polynomially solvable and that all exceptions can be characterized when is a semicomplete digraph, that is, a digraph with no pair of nonadjacent vertices. In this paper we extend these results to paths which are both edge‐disjoint and arc‐disjoint in semicomplete mixed graphs, that is, a mixed graph in which every pair of distinct vertices has either an arc, an edge, or both an arc and an edge between them. We give a complete characterization of the negative instances and explain how this gives rise to a polynomial algorithm for the problem.

A Comparative Study: Domination Parameters and Physico-Chemical Properties of Benzenoid Hydrocarbons

In graph theory, a graph is a mathematical structure that consists of a set of vertices and a set of edges connecting those pairs of vertices. Let G = ( V , E ) be a graph, a dominating set for G is a subset D ⊆ V of vertices such that every vertex in V ∖ D has at least one neighbor in D. The domination number of G, denoted by γ ( G ) , is the minimum cardinality of a dominating set in G. Recently, Khan (2023) performed a comparative study of seven important domination parameters with the π -electronic energy of benzenoid hydrocarbons, with a new theoretical approach. Physico-chemical properties are crucial for understanding the physical and chemical behavior of a molecule. In this paper, we perform a comparative study of those seven domination parameters with the physico-chemical properties of benzenoid hydrocarbons. We find the values of those domination parameters for the 22 lower benzenoid hydrocarbons and follow the statistical approach described by Khan (2023). The two physico-chemical properties to be chosen are the normal boiling point T b , a representative for intermolecular and van-der-Waals type interactions, whereas, the standard enthalpy/heat of formation Δ H f o is referred to represent the thermal properties of a molecular graph. This study further enhances that the performance of paired domination number is exceptional among all other parameters and highly correlated with both of the physico-chemical properties. Its correlation coefficient is 0.9981 (respectively, 0.9726) with T b (respectively, Δ H f o ). The outstanding correlation coefficient of paired domination number ensures further usage in quantitative structure-property relationship (QSPR) models.

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.

Extremal polyphenyl chains with respect to the Kirchhoff index

Let G be a simple connected graph. The Kirchhoff index of a graph is the sum of the resistance distance between all vertex pairs in the graph G. The resistance distance in a graph is equivalent to the effective resistance between any node pairs in the electrical network obtained by replacing each edge in the graph G with a unit resistance. The polyphenyl chain PL(S) is a graph with n hexagons where each hexagon can be reduced to a vertex and connect to a path. In this paper, we characterize polyphenyl chains with maximal and minimal Kirchhoff index.

Injective coloring of subclasses of chordal graphs

An injective k-coloring of a graph G=(V,E) is a function f:V→{1,2,…,k} such that for every pair of vertices u and v having a common neighbor, f(u)≠f(v). The injective chromatic number χi(G) of a graph G is the minimum k for which G admits an injective k-coloring. Given a graph G and a positive integer k, Decide Injective Coloring Problem is to decide whether G admits an injective k-coloring. Decide Injective Coloring Problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this result by proving that Decide Injective coloring Problem remains NP-complete for undirected path graphs, a proper subclass of chordal graphs. Moreover, we show that it is not possible to approximate the injective chromatic number of an undirected path graph within a factor of n1/3−ε in polynomial time for every ε>0 unless ZPP = NP. On the positive side, we prove that the injective chromatic number of an interval graph is either Δ(G) or Δ(G)+1, where Δ(G) is the maximum degree of G. We also characterize the interval graphs having χi(G)=Δ(G) and χi(G)=Δ(G)+1. As a consequence of this characterization, we obtain a linear time algorithm to find the injective chromatic number of an interval graph.

The λ -fold Spectrum Problem for the Orientations of the 6-Cycle

The λ-fold complete symmetric directed graph of order v, denoted λKv*, is the directed graph on v vertices and λ directed edges in each direction between each pair of vertices. For a given directed graph D, the set of all v for which λKv* admits a D-decomposition is called the λ-fold spectrum of D. In this paper, we settle the λ-fold spectrum of each of the nine non-isomorphic orientations of a 6-cycle.

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.

Extremal values on the Kirchhoff index of the line graph of trees

The computation of resistance distance and the Kirchhoff index is a classical problem that has been extensively investigated by numerous mathematicians, physicists, and scientists. Consider a simple connected graph G with vertex set V(G) and edge set E(G). If we replace each edge of G with a resistor of 1 ohm resistance, we create an electrical network N; in that case, the distance between any two nodes between network N is called resistance distance. The Kirchhoff index is a mathematical term that quantifies the complexity of a graph; it is defined as the total resistance distance among each pair of vertices in G. The line graph LG is constructed from G by swapping out the edges for vertices; in LG, if two vertices share endpoints in G, then they are connected in LG. In this study, extremal values for the Kirchhoff index of the line graph of trees are calculated. Further, we will also calculate the Kirchhoff index for the line graph of some special trees and establish the relationship between them.

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.

On a Norine–Thomas conjecture concerning minimal bricks

AbstractA 3‐connected graph is a brick if has a perfect matching, for each pair of vertices of . A brick is minimal if ceases to be a brick for every edge . Norine and Thomas proved that each minimal brick contains at least three vertices of degree three and made a stronger conjecture: there exists such that every minimal brick has at least cubic vertices. In this paper, we prove this conjecture holds for all minimal bricks of an average degree no less than 23/5. As its corollary, we show that each minimal brick on vertices contains more than vertices of degree at most four.

Optimal resistor networks

AbstractGiven a graph on vertices with edges, each of unit resistance, how small can the average resistance between pairs of vertices be? There are two very plausible extremal constructions — graphs like a star, and graphs which are close to regular — with the transition between them occurring when the average degree is 3. However, in this paper, we show that there are significantly better constructions for a range of average degree including average degree near 3. A key idea is to link this question to a analogous question about rooted graphs — namely ‘which rooted graph minimises the average resistance to the root?’. The rooted case is much simpler to analyse that the unrooted, and the one of the main results of this paper is that the two cases are asymptotically equivalent.

A Note on Wiener and Hyper-Wiener Indices of Abid-Waheed Graph

A Hosoya polynomial is a polynomial connected to a molecular graph, which is a graph representation of a chemical compound with atoms as vertices and chemical bonds as edges. A graph invariant is the Hosoya polynomial; it is a graph attribute that does not change under graph isomorphism. It provides information about the number of unique non-empty subgraphs in a given graph. A molecular graph's size and branching complexity are determined by a topological metric known as the Wiener index. The Wiener index of each pair of vertices in a molecular network is the sum of those distances. The topological index, one of the various classes of graph invariants, is a real number related to a connected graph's structure .The goal of this article is to compute the Hosoya polynomial of some class of Abid-Waheed graph. Further, this research focused on a C++ algorithm to calculate the wiener index of and . The Wiener index (W∗I) and Hyper-Wiener index (H∗W∗I) are calculated using Hosoya polynomial (H∗-polynomial) of some family of Abid-Waheed graphs and . Illustrations and applications are given to enhance the research work.

Asymptotic coarse Ricci curvature of inhomogeneous random graph

In this paper, we study the coarse Ricci curvature of inhomogeneous random graph on vertex set [n]≔{1,2,…,n}. In this graph, each pair of vertices (i,j) forms an edge independently with probability λnαgin,jn for some function g(x,y)∈(0,1], constants λ>0 and α∈[0,1]. We derive the asymptotic coarse Ricci curvature of this random graph. Phase transition phenomenon exists as α varies from zero to one.

Deterministic Replacement Path Covering

In this article, we provide a unified and simplified approach to derandomize central results in the area of fault-tolerant graph algorithms. Given a graph \(G\) , a vertex pair \((s,t)\in V(G)\times V(G)\) , and a set of edge faults \(F\subseteq E(G)\) , a replacement path \(P(s,t,F)\) is an \(s\) - \(t\) shortest path in \(G\setminus F\) . For integer parameters \(L,f\) , a replacement path covering ( \(\mathsf{RPC}\) ) is a collection of subgraphs of \(G\) , denoted by \(\mathcal{G}_{L,f}=\{G_{1},\ldots,G_{r}\}\) , such that for every set \(F\) of at most \(f\) faults (i.e., \(|F|\leq f\) ) and every replacement path \(P(s,t,F)\) of at most \(L\) edges, there exists a subgraph \(G_{i}\in\mathcal{G}_{L,f}\) that contains all the edges of \(P\) and does not contain any of the edges of \(F\) . The covering value of the \(\mathsf{RPC}\) \(\mathcal{G}_{L,f}\) is then defined to be the number of subgraphs in \(\mathcal{G}_{L,f}\) . In the randomized setting, it is easy to build an \((L,f)\) - \(\mathsf{RPC}\) with covering value of \(O(\max\{L,f\}^{\min\{L,f\}}\cdot\min\{L,f\}\cdot \log n)\) , but to this date, there is no efficient deterministic algorithm with matching bounds. As noted recently by Alon et al. (ICALP 2019), this poses the key barrier for derandomizing known constructions of distance sensitivity oracles and fault-tolerant spanners. We show the following: — There exist efficient deterministic constructions of \((L,f)\) - \(\mathsf{RPC}\) s whose covering values almost match the randomized ones, for a wide range of parameters. Our time and value bounds improve considerably over the previous construction of Parter (DISC 2019). Our algorithms are based on the introduction of a novel notion of hash families that we call HM hash families. We then show how to construct these hash families from (algebraic) error correcting codes such as Reed–Solomon codes and Algebraic-Geometric codes. — For every \(L,f\) , and \(n\) , there exists an \(n\) -vertex graph \(G\) whose \((L,f)\) - \(\mathsf{RPC}\) covering value is \(\Omega(L^{f})\) . This lower bound is obtained by exploiting connections to the problem of designing sparse fault-tolerant breadth first search (BFS) structures. An application of our above deterministic constructions is the derandomization of the algebraic construction of the distance sensitivity oracle by Weimann and Yuster (FOCS 2010). The preprocessing and query time of our deterministic algorithm nearly match the randomized bounds. This resolves the open problem of Alon et al. (ICALP 2019). Additionally, we show a derandomization of the randomized construction of vertex fault-tolerant spanners by Dinitz and Krauthgamer (PODC 2011) and Braunschvig et al. (Theor. Comput. Sci., 2015). The time complexity and the size bounds of the output spanners nearly match the randomized counterparts.

Radio labeling of biconvex split graphs

ABSTRACT A radio labeling of a graph G is a function f : V ( G ) → { 0 , 1 , … } such that for every pair of distinct vertices u , v ∈ V ( G ) , | f ( u ) − f ( v ) | ≥ 1 + diam ( G ) − d ( u , v ) , where diam(G) denotes the diameter of the graph and d(u, v) is the distance between the vertices u and v. The span of a radio labeling f of a graph G is the difference between the least and the largest labels assigned by f and is denoted by, span(f). The radio number of a graph G denoted by, rn(G), is the least positive integer s, such that there exists a radio labeling of G with span s. In this paper, we study the radio labeling of a special class of split graphs called biconvex split graphs of diameter three and we obtain both a lower bound and an upper bound for the radio number of biconvex split graphs of diameter three. Further, we determine the radio number of biconvex split graphs with three maximum degree vertices having disjoint independent neighbors.

Learning a Contact Potential Field for Modeling the Hand-Object Interaction.

Estimating and synthesizing the hand's manipulation of objects is central to understanding human behaviour. To accurately model the interaction between the hand and object (referred to as the "hand-object"), we must not only focus on the pose of the hand and object, but also consider the contact between them. This contact provides valuable information for generating semantically and physically plausible grasps. In this paper, we propose an explicit contact representation called Contact Potential Field (CPF). In CPF, we model the contact between a pair of hand-object vertices as a spring-mass system. This system encodes the distance of the pair, as well as a likelihood of that contact being stable. Therefore, the system of multiple extended and compressed springs forms an elastic potential field with minimal energy at the optimal grasp position. We apply CPF to two relevant tasks, namely, hand-object pose estimation and grasping pose generation. Extensive experiments on the two challenging tasks and three commonly used datasets have demonstrated that our method can achieve state-of-the-art in several reconstruction metrics, allowing us to produce more physically plausible hand-object poses even when the ground-truth exhibits severe interpenetration or disjointedness.

Investigating Metric Dimension and Edge Metric Dimension of Hexagonal Boron Nitride and Carbon Nanotubes

When there is a difference in the distance between two vertices in a simple linked graph, then a vertex $x$ resolves both $u$ and $v$. If at least one vertex in $S$ distinguishes each pair of distinct vertices in $G$, then a set $S$ of vertices in $G$ is referred to as a resolving set. $G$'s metric dimension is the minimum number of vertices required in a resolving set. A subset $S$ of vertices in a simple connected graph is called an edge metric generator if each vertex can tell any two distinct edges $e_1$ and $e_2$ apart by their respective distances from each other. The edge metric dimension (EMD), denoted as $\mathrm{dim}_e(G)$, is the smallest cardinality of such a subset $S$ that serves as an edge metric generator for $G$. The primary objective of this study is to investigate the edge metric dimension (EMD) of hexagonal boron nitride and carbon nanotube structures.

Cover and hitting times of hyperbolic random graphs

AbstractWe study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range . In particular, we first focus on the expected time for a random walk to hit a given vertex or visit, that is, cover, all vertices. We show that, a.a.s. (with respect to the HRG), and up to multiplicative constants: the cover time is , the maximum hitting time is , and the average hitting time is . We then determine the expected time to commute between two given vertices a.a.s., up to a small factor polylogarithmic in , and under some mild hypothesis on the pair of vertices involved. Our results are proved by controlling effective resistances using the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane, on which we overlay a forest‐like structure.