Including geometry in graph representations: A quadratic-time graph isomorphism algorithm and its applications

The most important step in the structural synthesis of planetary gear trains (PGTs) requires the identification of isomorphism (rotational as well as displacement) between the graphs which represent the kinematic structure of planetary gear train. Previously used methods for identifying graph isomorphism yielded incorrect results. Literature review in this area shows there is inconsistency in results from six link, one degree-of-freedom onwards. The purpose of this paper is to present an efficient methodology through the use of Loop concept and Hamming number concept to detect displacement and rotational isomorphism in PGTs in an unambiguous way. New invariants for rotational graphs and displacement graphs called geared chain hamming strings and geared chain loop hamming strings are developed respectively to identify rotational and displacement isomorphism. This paper also presents a procedure to redraw conventional graph representation that not only clarifies the kinematic structure of a PGT but also averts the problem of pseudo isomorphism. Finally a thorough analysis of existing methods is carried out using the proposed technique and the results in the category of six links one degree-of-freedom are established and an Atlas comprises of graph representations in conventional form as well as in new form is presented.

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The Decision-Making Neural Networks Model for Solving the Graph Isomorphism Problem

Chinese Journal of Computers ◽

10.3724/sp.j.1016.2010.00300 ◽

2010 ◽

Vol 33 (2) ◽

pp. 300-304 ◽

Cited By ~ 1

Author(s):

Jin-Hua NAN ◽

Huan QI

Keyword(s):

Neural Networks ◽

Decision Making ◽

Graph Isomorphism ◽

Isomorphism Problem ◽

Graph Isomorphism Problem

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Research on Blockchain-Based E-Bidding System

Applied Sciences ◽

10.3390/app11094011 ◽

2021 ◽

Vol 11 (9) ◽

pp. 4011

Author(s):

Dan Wang ◽

Jindong Zhao ◽

Chunxiao Mu

Keyword(s):

Data Transmission ◽

Privacy Protection ◽

Graph Isomorphism ◽

Third Party ◽

Business Logic ◽

Core Concept ◽

The Core ◽

Blockchain Technology ◽

Smart Contract ◽

Bidding Process

In the field of modern bidding, electronic bidding leads a new trend of development, convenience and efficiency and other significant advantages effectively promote the reform and innovation of China’s bidding field. Nowadays, most systems require a strong and trusted third party to guarantee the integrity and security of the system. However, with the development of blockchain technology and the rise of privacy protection, researchers has begun to emphasize the core concept of decentralization. This paper introduces a decentralized electronic bidding system based on blockchain and smart contract. The system uses blockchain to replace the traditional database and uses chaincode to process business logic. In data interaction, encryption techniques such as zero-knowledge proof based on graph isomorphism are used to improve privacy protection, which improves the anonymity of participants, the privacy of data transmission, and the traceability and verifiable of data. Compared with other electronic bidding systems, this system is more secure and efficient, and has the nature of anonymous operation, which fully protects the privacy information in the bidding process.

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Training Sensitivity in Graph Isomorphism Network

Proceedings of the 29th ACM International Conference on Information & Knowledge Management ◽

10.1145/3340531.3412089 ◽

2020 ◽

Author(s):

Md. Khaledur Rahman

Keyword(s):

Graph Isomorphism

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MET: a Java package for fast molecule equivalence testing

Journal of Cheminformatics ◽

10.1186/s13321-020-00480-1 ◽

2020 ◽

Vol 12 (1) ◽

Author(s):

Jördis-Ann Schüler ◽

Steffen Rechner ◽

Matthias Müller-Hannemann

Keyword(s):

Chemical Properties ◽

Graph Isomorphism ◽

Isomorphism Problem ◽

Equivalence Classes ◽

Equivalence Testing ◽

Equivalence Test ◽

Graph Isomorphism Problem ◽

2D Structure ◽

Node Labels ◽

Labelled Graphs

AbstractAn important task in cheminformatics is to test whether two molecules are equivalent with respect to their 2D structure. Mathematically, this amounts to solving the graph isomorphism problem for labelled graphs. In this paper, we present an approach which exploits chemical properties and the local neighbourhood of atoms to define highly distinctive node labels. These characteristic labels are the key for clever partitioning molecules into molecule equivalence classes and an effective equivalence test. Based on extensive computational experiments, we show that our algorithm is significantly faster than existing implementations within , and . We provide our Java implementation as an easy-to-use, open-source package (via GitHub) which is compatible with . It fully supports the distinction of different isotopes and molecules with radicals.

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Graph isomorphism restricted by lists

Theoretical Computer Science ◽

10.1016/j.tcs.2021.01.027 ◽

2021 ◽

Vol 860 ◽

pp. 51-71

Author(s):

Pavel Klavík ◽

Dušan Knop ◽

Peter Zeman

Keyword(s):

Graph Isomorphism

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Isomorphism, canonization, and definability for graphs of bounded rank width

Communications of the ACM ◽

10.1145/3453943 ◽

2021 ◽

Vol 64 (5) ◽

pp. 98-105

Author(s):

Martin Grohe ◽

Daniel Neuen

Keyword(s):

Fixed Point ◽

Complexity Theory ◽

Polynomial Time ◽

Graph Isomorphism ◽

Isomorphism Problem ◽

Descriptive Complexity ◽

Graph Isomorphism Problem ◽

Structural Graph ◽

Graph Classes ◽

Bounded Rank

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

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Completeness of an axiomatization of graph isomorphism via graph rewriting in Coq

Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs ◽

10.1145/3372885.3373831 ◽

2020 ◽

Cited By ~ 1

Author(s):

Christian Doczkal ◽

Damien Pous

Keyword(s):

Graph Isomorphism ◽

Graph Rewriting

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Applying graph database technology for analyzing perturbed co-expression networks in cancer

Database ◽

10.1093/database/baaa110 ◽

2020 ◽

Vol 2020 ◽

Author(s):

Claire M Simpson ◽

Florian Gnad

Keyword(s):

Relational Databases ◽

Molecular Mechanisms ◽

Biological Data ◽

Database Management System ◽

Graph Database ◽

Graph Databases ◽

Graph Representations ◽

Rnaseq Data ◽

Database Technology ◽

Speed Accuracy

Abstract Graph representations provide an elegant solution to capture and analyze complex molecular mechanisms in the cell. Co-expression networks are undirected graph representations of transcriptional co-behavior indicating (co-)regulations, functional modules or even physical interactions between the corresponding gene products. The growing avalanche of available RNA sequencing (RNAseq) data fuels the construction of such networks, which are usually stored in relational databases like most other biological data. Inferring linkage by recursive multiple-join statements, however, is computationally expensive and complex to design in relational databases. In contrast, graph databases store and represent complex interconnected data as nodes, edges and properties, making it fast and intuitive to query and analyze relationships. While graph-based database technologies are on their way from a fringe domain to going mainstream, there are only a few studies reporting their application to biological data. We used the graph database management system Neo4j to store and analyze co-expression networks derived from RNAseq data from The Cancer Genome Atlas. Comparing co-expression in tumors versus healthy tissues in six cancer types revealed significant perturbation tracing back to erroneous or rewired gene regulation. Applying centrality, community detection and pathfinding graph algorithms uncovered the destruction or creation of central nodes, modules and relationships in co-expression networks of tumors. Given the speed, accuracy and straightforwardness of managing these densely connected networks, we conclude that graph databases are ready for entering the arena of biological data.

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