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Formulating and Solving Linear Programs
Published in Craig A. Tovey, Linear Optimization and Duality, 2020
Question: Why must this constraint be written for each ordered pair, not just for each unordered pair?14Question: There are (|J|)(|J|−1) balancing constraints. If the balancing requirement required exactly the same proportion of weight capacity in each compartment, how many constraints would be needed?15
Chapter 11: Miscellaneous Topics Used for Engineering Problems
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
A linear graph (or simply a graph) G = (V, E) consists of a set of objects V = {v1, v2, …} called vertices, and another set E = {e1, e2, …}, whose elements are called edges, such that each edge ek is identified with an unordered pair (vi, vj) of vertices. The vertices (vi, vj) associated with edge ek are called the end vertices of ek.
Introduction to Graph Models
Published in Jonathan L. Gross, Jay Yellen, Mark Anderson, Graph Theory and Its Applications, 2018
Jonathan L. Gross, Jay Yellen, Mark Anderson
The ordered-pair representation could also prove awkward in implementing algorithms for which the graphs or digraphs are dynamic structures (i.e., they change during the algorithm). Whenever the direction on a particular edge must be reversed, the associated ordered pair has to be deleted and replaced by its reverse. Even worse, if a directed edge is to become undirected, then an ordered pair must be replaced with an unordered pair. Similarly, the undirected and directed edges of a mixed graph would require two different types of objects.
Application of approximate concept and graph theory to beam and plate topology optimization
Published in Engineering Optimization, 2023
Shuanjun Liu, Hai Huang, Jianhongyu Li, Yipeng Zhang, Jiayi Fu
Based on the set description of the graph, this work uses an ergodic method to realize the program design of topology rationality. One unordered pair in is selected as the two initial elements of the connected domain set , and the remaining unordered pairs in are repeatedly traversed and compared with the elements in . If contains points in the unordered pair, the unordered pair is added to and marked as ‘joined’ (If no point joins the connected domain in a traversal, this indicates that the graph is not connected) and continue until all disordered pairs in are marked as ‘joined’. If , the algorithm's time complexity is .
An energy-based analysis of reduced-order models of (networked) synchronous machines
Published in Mathematical and Computer Modelling of Dynamical Systems, 2019
T. W. Stegink, C. De Persis, A. J. Van Der Schaft
Consider a power grid consisting of buses. The network is represented by a connected and undirected graph , where the set of nodes, , is the set of buses representing the synchronous machines and the set of edges, , is the set of transmission lines connecting the buses where each edge is an unordered pair of two vertices . Given a node , then the set of neighbouring nodes is denoted by .
Dombi Fuzzy Graphs
Published in Fuzzy Information and Engineering, 2018
S. Ashraf, S. Naz, E. E. Kerre
A graph is a mathematical structure consisting of a set of vertices and a set of edges , where each edge is an unordered pair of distinct vertices. Two vertices x and y of a graph G are adjacent if . A vertex joined by an edge to a vertex x is called a neighbor of x. The number of edges joined with a vertex x of a graph G is called the degree of x in G denoted by deg or deg. A graph with no loops and multiple edges is called simple graph. Throughout this paper, we will consider only undirected, simple graphs. The complement of a graph G, is a graph having vertex set same as in G, in which two vertices are adjacent if and only if they are not adjacent in G. If there exists a one-one correspondence between the vertices of two graphs and which preserves adjacency, then the graphs and are called isomorphic.