B. Isomorphic Graphs

2020 ◽  
pp. 21-24
Keyword(s):  
Isomorphic Graphs

Topological Synthesis of Epicyclic Gear Trains Using Vertex Incidence Polynomial

2017 ◽  
Vol 139 (6) ◽  
Author(s):  
Vinjamuri Venkata Kamesh ◽  
Kuchibhotla Mallikarjuna Rao ◽  
Annambhotla Balaji Srinivasa Rao
Keyword(s):  
High Velocity ◽  
Mechanical Energy ◽  
Gear Train ◽  
Recursive Method ◽  
Gear Pair ◽  
Simple Method ◽  
Epicyclic Gear ◽  
Thick Line ◽  
Gear Trains ◽  
Isomorphic Graphs

Epicyclic gear trains (EGTs) are used in the mechanical energy transmission systems where high velocity ratios are needed in a compact space. It is necessary to eliminate duplicate structures in the initial stages of enumeration. In this paper, a novel and simple method is proposed using a parameter, Vertex Incidence Polynomial (VIP), to synthesize epicyclic gear trains up to six links eliminating all isomorphic gear trains. Each epicyclic gear train is represented as a graph by denoting gear pair with thick line and transfer pair with thin line. All the permissible graphs of epicyclic gear trains from the fundamental principles are generated by the recursive method. Isomorphic graphs are identified by calculating VIP. Another parameter “Rotation Index” (RI) is proposed to detect rotational isomorphism. It is found that there are six nonisomorphic rotation graphs for five-link one degree-of-freedom (1-DOF) and 26 graphs for six-link 1-DOF EGTs from which all the nonisomorphic displacement graphs can be derived by adding the transfer vertices for each combination. The proposed method proved to be successful in clustering all the isomorphic structures into a group, which in turn checked for rotational isomorphism. This method is very easy to understand and allows performing isomorphism test in epicyclic gear trains.

The Degree Code—A New Mechanism Identifier

1993 ◽  
Vol 115 (3) ◽  
pp. 627-630 ◽  
Author(s):  
C. S. Tang ◽  
Tyng Liu
Keyword(s):  
Storage System ◽  
Graph Isomorphism ◽  
Critical Issue ◽  
Mathematical Representation ◽  
Structural Synthesis ◽  
Synthesis Of Mechanisms ◽  
Code Algorithm ◽  
Isomorphic Graphs ◽  
And Storage ◽  
Isomorphism Identification

An important step in the structural synthesis of mechanisms requires the identification of isomorphism between the graphs which represents the mechanism topology. Previously used methods for identifying graph isomorphism either yield incorrect results for some cases or their algorithms are computationally inefficient for this application. This paper describes a new isomorphism identification method which is well suited for the automated structural synthesis of mechanisms. This method uses a new and compact mathematical representation for a graph, called the Degree Code, to identify graph isomorphism. Isomorphic graphs have identical Degree Codes; nonisomorphic graphs have distinct Degree Codes. Therefore, by examining the Degree Codes of the graphs, graph isomorphism is easily and correctly identified. This Degree Code algorithm is simpler and more efficient than other methods for identifying isomorphism correctly. In addition, the Degree Code can serve as an effective nomenclature and storage system for graphs or mechanisms. Although this identification scheme was developed specifically for the structural synthesis of mechanisms, it can be applied to any area where graph isomorphism is a critical issue.

scholarly journals Neighborhood Reconstruction and Cancellation of Graphs

10.37236/6676 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Richard H. Hammack ◽  
Cristina Mullican
Keyword(s):  
Direct Product ◽  
Structural Characterization ◽  
Unsolved Problem ◽  
Reconstruction Problem ◽  
Cancellation Problem ◽  
Open Vertex ◽  
Isomorphic Graphs ◽  
Product Of Graphs

We connect two seemingly unrelated problems in graph theory.Any graph $G$ has a neighborhood multiset $\mathscr{N}(G)= \{N(x) \mid x\in V(G)\}$ whose elements are precisely the open vertex-neighborhoods of $G$. In general there exist non-isomorphic graphs $G$ and $H$ for which $\mathscr{N}(G)=\mathscr{N}(H)$. The neighborhood reconstruction problem asks the conditions under which $G$ is uniquely reconstructible from its neighborhood multiset, that is, the conditions under which $\mathscr{N}(G)=\mathscr{N}(H)$ implies $G\cong H$. Such a graph is said to be neighborhood-reconstructible.The cancellation problem for the direct product of graphs seeks the conditions under which $G\times K\cong H\times K$ implies $G\cong H$. Lovász proved that this is indeed the case if $K$ is not bipartite. A second instance of the cancellation problem asks for conditions on $G$ that assure $G\times K\cong H\times K$ implies $G\cong H$ for any bipartite~$K$ with $E(K)\neq \emptyset$. A graph $G$ for which this is true is called a cancellation graph.We prove that the neighborhood-reconstructible graphs are precisely the cancellation graphs. We also present some new results on cancellation graphs, which have corresponding implications for neighborhood reconstruction. We are particularly interested in the (yet-unsolved) problem of finding a simple structural characterization of cancellation graphs (equivalently, neighborhood-reconstructible graphs).

Classifying Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Graph Theory ◽  
Rational Number ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Reconstruction Problem ◽  
Regular Polyhedra ◽  
Four Color Theorem ◽  
Stanislaw Ulam ◽  
Irregular Graphs ◽  
Isomorphic Graphs

This chapter considers the richness of mathematics and mathematicians' responses to it, with a particular focus on various types of graphs. It begins with a discussion of theorems from many areas of mathematics that have been judged among the most beautiful, including the Euler Polyhedron Formula; the number of primes is infinite; there are five regular polyhedra; there is no rational number whose square is 2; and the Four Color Theorem. The chapter proceeds by describing regular graphs, irregular graphs, irregular multigraphs and weighted graphs, subgraphs, and isomorphic graphs. It also analyzes the degrees of the vertices of a graph, along with concepts and ideas concerning the structure of graphs. Finally, it revisits a rather mysterious problem in graph theory, introduced by Stanislaw Ulam and Paul J. Kelly, that no one has been able to solve: the Reconstruction Problem.

scholarly journals Enumerating Tree-Like Graphs and Polymer Topologies with a Given Cycle Rank

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1295
Author(s):  
Naveed Ahmed Azam ◽  
Aleksandar Shurbevski ◽  
Hiroshi Nagamochi
Keyword(s):  
Chemical Compounds ◽  
Canonical Representation ◽  
Cycle Rank ◽  
Multiple Edges ◽  
Isomorphic Graphs ◽  
Important Notion

Cycle rank is an important notion that is widely used to classify, understand, and discover new chemical compounds. We propose a method to enumerate all non-isomorphic tree-like graphs of a given cycle rank with self-loops and no multiple edges. To achieve this, we develop an algorithm to enumerate all non-isomorphic rooted graphs with the required constraints. The idea of our method is to define a canonical representation of rooted graphs and enumerate all non-isomorphic graphs by generating the canonical representation of rooted graphs. An important feature of our method is that for an integer n≥1, it generates all required graphs with n vertices in O(n) time per graph and O(n) space in total, without generating invalid intermediate structures. We performed some experiments to enumerate graphs with a given cycle rank from which it is evident that our method is efficient. As an application of our method, we can generate tree-like polymer topologies of a given cycle rank with self-loops and no multiple edges.

INDUCED SUBGRAPHS OF GAMMA GRAPHS

2013 ◽  
Vol 05 (03) ◽  
pp. 1350012 ◽  
Author(s):  
N. SRIDHARAN ◽  
S. AMUTHA ◽  
S. B. RAO
Keyword(s):  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Vertex Set ◽  
Isomorphic Graphs

Let G be a graph. The gamma graph of G denoted by γ ⋅ G is the graph with vertex set V(γ ⋅ G) as the set of all γ-sets of G and two vertices D and S of γ ⋅ G are adjacent if and only if |D ∩ S| = γ(G) – 1. A graph H is said to be a γ-graph if there exists a graph G such that γ ⋅ G is isomorphic to H. In this paper, we show that every induced subgraph of a γ-graph is also a γ-graph. Furthermore, if we prove that H is a γ-graph, then there exists a sequence {Gn} of non-isomorphic graphs such that H = γ ⋅ Gn for every n.

On the Hamilton Cycle of the Hypercube

2011 ◽  
Vol 480-481 ◽  
pp. 922-927 ◽  
Author(s):  
Yan Zhong Hu ◽  
Hua Dong Wang
Keyword(s):  
Euler Characteristic ◽  
Interconnection Networks ◽  
Cartesian Product ◽  
Hamilton Cycle ◽  
The Other ◽  
Product Graph ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Isomorphic Graphs ◽  
The Relationship

Hypercube is one of the basic types of interconnection networks. In this paper, we use the concept of the Cartesian product graph to define the hypercube Qn, we study the relationship between the isomorphic graphs and the Cartesian product graphs, and we get the result that there exists a Hamilton cycle in the hypercube Qn. Meanwhile, the other properties of the hypercube Qn, such as Euler characteristic and bipartite characteristic are also introduced.

scholarly journals Structural Synthesis of Articulated Manipulators with Non-Fractionated or Fractionated Kinematic Chains without Isomorphism

2021 ◽  
Vol 11 (20) ◽  
pp. 9658
Author(s):  
Ho Sung Park ◽  
Jae Kyung Shim ◽  
Woon Ryong Kim ◽  
Tae Woong Yun
Keyword(s):  
Structural Characteristics ◽  
Kinematic Chain ◽  
Kinematic Structure ◽  
Free Graph ◽  
Kinematic Chains ◽  
Conceptual Design Phase ◽  
Fixed Base ◽  
Force Transmissibility ◽  
Isomorphic Graphs ◽  
Two Degree Of Freedom

As the kinematic structure of an articulated manipulator affects the characteristics of its motion, rigidity, vibration, and force transmissibility, finding the most suitable kinematic structure for the desired task is important in the conceptual design phase. This paper proposes a systematic method for generating non-isomorphic graphs of articulated manipulators that consist of a fixed base, an end-effector, and a two-degree-of-freedom (DOF) intermediate kinematic chain connecting the two. Based on the analysis of the structural characteristics of articulated manipulators, the conditions that must be satisfied for manipulators to have a desired DOF is identified. Then, isomorphism-free graph generation methods are proposed based on the concepts of the symmetry of a graph, and the number of graphs generated are determined. As a result, 969 graphs of articulated manipulators that have two-DOF non-fractionated intermediate kinematic chains and 33,438 graphs with two-DOF fractionated intermediate kinematic chains are generated, including practical articulated manipulators widely used in industry.

scholarly journals Graph Deformer Network

2021 ◽  
Author(s):  
Wenting Zhao ◽  
Yuan Fang ◽  
Zhen Cui ◽  
Tong Zhang ◽  
Jian Yang
Keyword(s):  
Receptive Fields ◽  
Graph Isomorphism ◽  
Feature Space ◽  
Virtual Space ◽  
Learning Ability ◽  
Potential Applications ◽  
Anchor Nodes ◽  
Structural Loss ◽  
Irregular Data ◽  
Isomorphic Graphs

Convolution learning on graphs draws increasing attention recently due to its potential applications to a large amount of irregular data. Most graph convolution methods leverage the plain summation/average aggregation to avoid the discrepancy of responses from isomorphic graphs. However, such an extreme collapsing way would result in a structural loss and signal entanglement of nodes, which further cause the degradation of the learning ability. In this paper, we propose a simple yet effective Graph Deformer Network (GDN) to fulfill anisotropic convolution filtering on graphs, analogous to the standard convolution operation on images. Local neighborhood subgraphs (acting like receptive fields) with different structures are deformed into a unified virtual space, coordinated by several anchor nodes. In the deformation process, we transfer components of nodes therein into affinitive anchors by learning their correlations, and build a multi-granularity feature space calibrated with anchors. Anisotropic convolutional kernels can be further performed over the anchor-coordinated space to well encode local variations of receptive fields. By parameterizing anchors and stacking coarsening layers, we build a graph deformer network in an end-to-end fashion. Theoretical analysis indicates its connection to previous work and shows the promising property of graph isomorphism testing. Extensive experiments on widely-used datasets validate the effectiveness of GDN in graph and node classifications.

scholarly journals Hearing the Shape of the Ising Model with a Programmable Superconducting-Flux Annealer

2014 ◽  
Vol 4 (1) ◽  
Author(s):  
Walter Vinci ◽  
Klas Markström ◽  
Sergio Boixo ◽  
Aidan Roy ◽  
Federico M. Spedalieri ◽  
...  
Keyword(s):  
Energy Spectrum ◽  
Ising Model ◽  
Partition Function ◽  
Low Energy ◽  
Partition Functions ◽  
Spectral Invariants ◽  
Physical Systems ◽  
Energy States ◽  
Isomorphic Graphs ◽  
Quantum Spectra

Abstract Two objects can be distinguished if they have different measurable properties. Thus, distinguishability depends on the Physics of the objects. In considering graphs, we revisit the Ising model as a framework to define physically meaningful spectral invariants. In this context, we introduce a family of refinements of the classical spectrum and consider the quantum partition function. We demonstrate that the energy spectrum of the quantum Ising Hamiltonian is a stronger invariant than the classical one without refinements. For the purpose of implementing the related physical systems, we perform experiments on a programmable annealer with superconducting flux technology. Departing from the paradigm of adiabatic computation, we take advantage of a noisy evolution of the device to generate statistics of low energy states. The graphs considered in the experiments have the same classical partition functions, but different quantum spectra. The data obtained from the annealer distinguish non-isomorphic graphs via information contained in the classical refinements of the functions but not via the differences in the quantum spectra.

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