Line graph theory reveals hidden spin frustration and bond frustration in molecular crystals with strong isotropy

2022 ◽  
Author(s):  
Rie Suizu ◽  
Kunio Awaga
Keyword(s):  
Graph Theory ◽  
Line Graph ◽  
Molecular Crystals ◽  
Line Graphs ◽  
Spin Frustration

Graph theory has demonstrated that only three lattices, namely, honeycomb, K4, and diamond lattices, possess a strong isotropic property. It is also recognized that their line graphs correspond to kagome,...

Logical Foundations of Kinematic Chains: Graphs, Line Graphs, and Hypergraphs

1990 ◽  
Vol 112 (1) ◽  
pp. 79-83 ◽  
Author(s):  
Frank Harary ◽  
Hong-Sen Yan
Keyword(s):  
Graph Theory ◽  
Line Graph ◽  
Kinematic Chain ◽  
Line Graphs ◽  
Kinematic Chains ◽  
Logical Foundations ◽  
Unique Graph ◽  
Theory Of Graphs ◽  
Admissible Graph

In terms of concepts from the theory of graphs and hypergraphs we formulate a precise structural characterization of a kinematic chain. To do this, we require the operations of line graph, intersection graph, and hypergraph duality. Using these we develop simple algorithms for constructing the unique graph G (KC) of a kinematic chain KC and (given an admissible graph G) for forming the unique kinematic chain whose graph is G. This one-to-one correspondence between kinematic chains and a class of graphs enables the mathematical and logical power, precision, concepts, and theorems of graph theory to be applied to gain new insights into the structure of kinematic chains.

scholarly journals The competition number of a generalized line graph is at most two

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Boram Park ◽  
Yoshio Sano
Keyword(s):  
Graph Theory ◽  
Necessary Conditions ◽  
Line Graph ◽  
Sufficient Conditions ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
International Audience ◽  
Necessary And Sufficient ◽  
Competition Number

Graph Theory International audience In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.

An Algorithm for Automatic Sketching of Planar Kinematic Chains

1985 ◽  
Vol 107 (1) ◽  
pp. 106-111 ◽  
Author(s):  
D. G. Olson ◽  
T. R. Thompson ◽  
D. R. Riley ◽  
A. G. Erdman
Keyword(s):  
Graph Theory ◽  
Line Graph ◽  
Kinematic Chain ◽  
Graph Representation ◽  
Synthesis Process ◽  
Line Graphs ◽  
Type Synthesis ◽  
Kinematic Chains ◽  
Synthesis Of Mechanisms ◽  
Computer Aided

One of the problems encountered in attempting to computerize type synthesis of mechanisms is that of automatically generating a computer graphics display of candidate kinematic chains or mechanisms. This paper presents the development of a computer algorithm for automatic sketching of kinematic chains as part of the computer-aided type synthesis process. Utilizing concepts from graph theory, it can be shown that a sketch of a kinematic chain can be obtained from its graph representation by simply transforming the graph into its line graph, and then sketching the line graph. The fundamentals of graph theory as they relate to the study of mechanisms are reviewed. Some new observations are made relating to graphs and their corresponding line graphs, and a novel procedure for transforming the graph into its line graph is presented. This is the basis of a sketching algorithm which is illustrated by computer-generated examples.

scholarly journals Computing Topological Indices for Para-Line Graphs of Anthracene

2019 ◽  
Vol 17 (1) ◽  
pp. 955-962 ◽  
Author(s):  
Zhiqiang Zhang ◽  
Zeshan Saleem Mufti ◽  
Muhammad Faisal Nadeem ◽  
Zaheer Ahmad ◽  
Muhammad Kamran Siddiqui ◽  
...  
Keyword(s):  
Graph Theory ◽  
Chemical Compound ◽  
Line Graph ◽  
Molecular Graph ◽  
Chemical Properties ◽  
Molar Refraction ◽  
Chromatographic Behavior ◽  
Chemical Graph ◽  
Chemical Graph Theory ◽  
And Mathematics

AbstractAtoms displayed as vertices and bonds can be shown by edges on a molecular graph. For such graphs we can find the indices showing their bioactivity as well as their physio-chemical properties such as the molar refraction, molar volume, chromatographic behavior, heat of atomization, heat of vaporization, magnetic susceptibility, and the partition coefficient. Today, industry is flourishing because of the interdisciplinary study of different disciplines. This provides a way to understand the application of different disciplines. Chemical graph theory is a mixture of chemistry and mathematics, which plays an important role in chemical graph theory. Chemistry provides a chemical compound, and graph theory transforms this chemical compound into a molecular graphwhich further is studied by different aspects such as topological indices.We will investigate some indices of the line graph of the subdivided graph (para-line graph) of linear-[s] Anthracene and multiple Anthracene.

scholarly journals Topological invariants for the line graphs of some classes of graphs

2019 ◽  
Vol 17 (1) ◽  
pp. 1483-1490
Author(s):  
Xiaoqing Zhou ◽  
Mustafa Habib ◽  
Tariq Javeed Zia ◽  
Asim Naseem ◽  
Anila Hanif ◽  
...  
Keyword(s):  
Communication Theory ◽  
Chemical Reactivity ◽  
Line Graph ◽  
Chemical Properties ◽  
Topological Invariants ◽  
Line Graphs ◽  
Physical And Chemical Properties ◽  
Physical And Chemical ◽  
Ladder Graphs ◽  
And Chemical Properties

AbstractGraph theory plays important roles in the fields of electronic and electrical engineering. For example, it is critical in signal processing, networking, communication theory, and many other important topics. A topological index (TI) is a real number attached to graph networks and correlates the chemical networks with physical and chemical properties, as well as with chemical reactivity. In this paper, our aim is to compute degree-dependent TIs for the line graph of the Wheel and Ladder graphs. To perform these computations, we first computed M-polynomials and then from the M-polynomials we recovered nine degree-dependent TIs for the line graph of the Wheel and Ladder graphs.

scholarly journals A Strengthening of Brooks' Theorem for Line Graphs

10.37236/632 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Clique Number ◽  
Line Graphs ◽  
Delta G

We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively. Thus Brooks' Theorem holds for line graphs of multigraphs in much stronger form. Using similar methods we then prove that if $G$ is the line graph of a multigraph with $\chi(G) \geq \Delta(G) \geq 9$, then $G$ contains a clique on $\Delta(G)$ vertices. Thus the Borodin-Kostochka Conjecture holds for line graphs of multigraphs.

On Spanning and Dominating Circuits in Graphs

1977 ◽  
Vol 20 (2) ◽  
pp. 215-220 ◽  
Author(s):  
L. Lesniak-Foster ◽  
James E. Williamson
Keyword(s):  
Line Graph ◽  
Dominating Set ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Vertex Set ◽  
On Line ◽  
One To One

AbstractA set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.

scholarly journals Efficient Algorithm for Generating Maximal L-Reflexive Trees

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 809
Author(s):  
Milica Anđelić ◽  
Dejan Živković
Keyword(s):  
Efficient Algorithm ◽  
Line Graph ◽  
Common Vertex ◽  
Line Graphs ◽  
Algebraic Numbers ◽  
Pisot Numbers ◽  
Full Characterization ◽  
Vertex Set ◽  
Second Largest Eigenvalue

The line graph of a graph G is another graph of which the vertex set corresponds to the edge set of G, and two vertices of the line graph of G are adjacent if the corresponding edges in G share a common vertex. A graph is reflexive if the second-largest eigenvalue of its adjacency matrix is no greater than 2. Reflexive graphs give combinatorial ground to generate two classes of algebraic numbers, Salem and Pisot numbers. The difficult question of identifying those graphs whose line graphs are reflexive (called L-reflexive graphs) is naturally attacked by first answering this question for trees. Even then, however, an elegant full characterization of reflexive line graphs of trees has proved to be quite formidable. In this paper, we present an efficient algorithm for the exhaustive generation of maximal L-reflexive trees.

Binomial incidence matrix of a semigraph

2020 ◽  
pp. 2150017
Author(s):  
Jyoti Shetty ◽  
G. Sudhakara
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Incidence Matrix ◽  
Systematic Approach ◽  
Line Graph ◽  
The Matrix ◽  
Theory Of Graphs

A semigraph, defined as a generalization of graph by  Sampathkumar, allows an edge to have more than two vertices. The idea of multiple vertices on edges gives rise to multiplicity in every concept in the theory of graphs when generalized to semigraphs. In this paper, we define a representing matrix of a semigraph [Formula: see text] and call it binomial incidence matrix of the semigraph [Formula: see text]. This matrix, which becomes the well-known incidence matrix when the semigraph is a graph, represents the semigraph uniquely, up to isomorphism. We characterize this matrix and derive some results on the rank of the matrix. We also show that a matrix derived from the binomial incidence matrix satisfies a result in graph theory which relates incidence matrix of a graph and adjacency matrix of its line graph. We extend the concept of “twin vertices” in the theory of graphs to semigraph theory, and characterize them. Finally, we derive a systematic approach to show that the binomial incidence matrix of any semigraph on [Formula: see text] vertices can be obtained from the incidence matrix of the complete graph [Formula: see text].

Motion Comparison Method Combining Segmented Multi-Joint Line Graphs with the SIFT Matching Technique

2014 ◽  
Vol 599-601 ◽  
pp. 1566-1570
Author(s):  
Ming Zeng ◽  
Hong Lin Ren ◽  
Qing Hao Meng ◽  
Chang Wei Chen ◽  
Shu Gen Ma
Keyword(s):  
Feature Matching ◽  
Line Graph ◽  
Comparison Method ◽  
Joint Line ◽  
Line Graphs ◽  
3D Motion ◽  
Motion Data ◽  
Comparison Results ◽  
Matching Technique ◽  
Sift Features

In this paper, an effective motion comparison method based on segmented multi-joint line graphs combined with the SIFT feature matching method is proposed. Firstly, the multi-joint 3D motion data are captured using the Kinect. Secondly, 3D motion data are normalized and distortion data are removed. Therefore, a 2D line graph can be obtained. Next, SIFT features of the 2D motion line graph are extracted. Finally, the line graphs are divided into several regions and then the comparison results can be calculated based on SIFT matching ratios between the tutor’s local line graph and the trainee’s local line graph. The experimental results show that the proposed method not only can easily deal with the several challenge problems in motion analysis, e.g., the problem of different rhythm of motions, the problem of a large amount of data, but also can provide detailed error correction cues.

Sign in / Sign up

Export Citation Format

Share Document