Homology of the line graph and of related graph-valued functions

1991 ◽  
Vol 56 (4) ◽  
pp. 400-404 ◽  
Author(s):  
Erich Prisner
Keyword(s):  
Line Graph ◽  
Related Graph

Fuzzy Labeling on Cycle Related Graph

2019 ◽  
Vol 7 (6) ◽  
pp. 1192-1194
Author(s):  
M.K. Pandurangan ◽  
T. Bharathi ◽  
S.Antony Vinoth
Keyword(s):  
Related Graph

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 A graph-based algorithm for detecting rigid domains in protein structures

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Truong Khanh Linh Dang ◽  
Thach Nguyen ◽  
Michael Habeck ◽  
Mehmet Gültas ◽  
Stephan Waack
Keyword(s):  
Structural Changes ◽  
Viterbi Algorithm ◽  
Structural Information ◽  
Line Graph ◽  
Clustering Algorithms ◽  
Protein Structures ◽  
Alternative Methods ◽  
Structural Transitions ◽  
Tuning Parameters ◽  
Conformational Ensembles

Abstract Background Conformational transitions are implicated in the biological function of many proteins. Structural changes in proteins can be described approximately as the relative movement of rigid domains against each other. Despite previous efforts, there is a need to develop new domain segmentation algorithms that are capable of analysing the entire structure database efficiently and do not require the choice of protein-dependent tuning parameters such as the number of rigid domains. Results We develop a graph-based method for detecting rigid domains in proteins. Structural information from multiple conformational states is represented by a graph whose nodes correspond to amino acids. Graph clustering algorithms allow us to reduce the graph and run the Viterbi algorithm on the associated line graph to obtain a segmentation of the input structures into rigid domains. In contrast to many alternative methods, our approach does not require knowledge about the number of rigid domains. Moreover, we identified default values for the algorithmic parameters that are suitable for a large number of conformational ensembles. We test our algorithm on examples from the DynDom database and illustrate our method on various challenging systems whose structural transitions have been studied extensively. Conclusions The results strongly suggest that our graph-based algorithm forms a novel framework to characterize structural transitions in proteins via detecting their rigid domains. The web server is available at http://azifi.tz.agrar.uni-goettingen.de/webservice/.

scholarly journals A New Method for Type Synthesis of 2R1T and 2T1R 3-DOF Redundant Actuated Parallel Mechanisms with Closed Loop Units

2020 ◽  
Vol 33 (1) ◽  
Author(s):  
Yongquan Li ◽  
Yang Zhang ◽  
Lijie Zhang
Keyword(s):  
Closed Loop ◽  
Screw Theory ◽  
Line Graph ◽  
Degree Of Freedom ◽  
Parallel Mechanisms ◽  
Type Synthesis ◽  
Allocation Criteria ◽  
Grassmann Line Geometry ◽  
Moving Platform ◽  
Atlas Method

Abstract The current type synthesis of the redundant actuated parallel mechanisms is adding active-actuated kinematic branches on the basis of the traditional parallel mechanisms, or using screw theory to perform multiple getting intersection and union to complete type synthesis. The number of redundant parallel mechanisms obtained by these two methods is limited. In this paper, based on Grassmann line geometry and Atlas method, a novel and effective method for type synthesis of redundant actuated parallel mechanisms (PMs) with closed-loop units is proposed. Firstly, the degree of freedom (DOF) and constraint line graph of the moving platform are determined successively, and redundant lines are added in constraint line graph to obtain the redundant constraint line graph and their equivalent line graph, and a branch constraint allocation scheme is formulated based on the allocation criteria. Secondly, a scheme is selected and redundant lines are added in the branch chains DOF graph to construct the redundant actuated branch chains with closed-loop units. Finally, the branch chains that meet the requirements of branch chains configuration criteria and F&C (degree of freedom & constraint) line graph are assembled. In this paper, two types of 2 rotational and 1 translational (2R1T) redundant actuated parallel mechanisms and one type of 2 translational and 1 rotational (2T1R) redundant actuated parallel mechanisms with few branches and closed-loop units were taken as examples, and 238, 92 and 15 new configurations were synthesized. All the mechanisms contain closed-loop units, and the mechanisms and the actuators both have good symmetry. Therefore, all the mechanisms have excellent comprehensive performance, in which the two rotational DOFs of the moving platform of 2R1T redundant actuated parallel mechanism can be independently controlled. The instantaneous analysis shows that all mechanisms are not instantaneous, which proves the feasibility and practicability of the method.

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.

ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS

2015 ◽  
Vol 91 (3) ◽  
pp. 353-367 ◽  
Author(s):  
JING HUANG ◽  
SHUCHAO LI
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Regular Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Subdivision Graph ◽  
Number Of Spanning Trees ◽  
Sharp Lower Bound

Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).

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