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1. • • A non-trivial graph is any graph that contains more than one vertex. In other words, it consists of at least two distinct points (vertices) that may or may not be connected by edges. Non-trivial graphs exhibit a more complex structure with a higher degree of connectivity between their vertices compared to trivial graphs.
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     Trivial Graph in Graph Theory

2. Top results related to what is a non-trivial graph theory

   • What is a non-trivial constructor in C++?

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     Answered Oct 10, 2010 · 104 votes

     In simple words a "trivial" special member function literally means a member function that does its job in a very straightforward manner. The "straightforward manner" means different thing for different kinds of special member functions.

     For a default constructor and destructor being "trivial" means literally "do nothing at all". For copy-constructor and copy-assignment operator, being "trivial" means literally "be equivalent to simple raw memory copying" (like copy with memcpy).

     If you define a constructor yourself, it is considered non-trivial, even if it doesn't do anything, so a trivial constructor must be implicitly defined by the compiler.

     In order for a special member function to satisfy the above requirements, the class must have a very simplistic structure, it must not require any hidden initializations when an object is being created or destroyed, or any hidden additional internal manipulations when it is being copied.

     For example, if class has virtual functions, it will require some extra hidden initializations when objects of this class are being created (initialize virtual method table and such), so the constructor for this class will not qualify as trivial.

     For another example, if a class has virtual base classes, then each object of this class might contain hidden pointers that point to other parts of the very same object. Such a self-referential object cannot be copied by a simple raw memory copy routine (like memcpy). Extra manipulations will be necessary to properly re-initialize the hidden pointers in the copy. For this reason the copy constructor and copy-assignment operator for this class will not qualify as trivial.

     For obvious reasons, this requirement is recursive: all subobjects of the class (bases and non-static members) must also have trivial constructors.

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   • what is the difference between trivial and non trivial objects

     Top Answer

     Answered Apr 20, 2020 · 7 votes

     The official definition of a trivial type can be found here.

     In simpler terms, a trivial type is either a fundamental type (int, float, etc.) or a type composed of only other trivial types, and without any of the special member functions listed here. Other member functions don't play a role.

     The point of triviality is that the type can be treated exactly like a fundamental type, in that objects of the type can be copied and moved with memcpy and constructed destructed without doing anything. Hence, triviality requires a type be essentially made only of fundamental types. This is what makes the copy, move, construction, and destruction operations relevant to the definition of trivial types. Other member functions don't play a role in triviality just as you can write void fn( int*, OtherArgs... ) without affecting whether or not an int is trivial, because you can think of member functions of T as essentially being free functions with the signature ReturnType member_function( T*, OtherArgs... ) that the compiler let's you call with the syntax a.member_function( other_args... ).

     As for alignment, it simply isn't all that relevant because it's all taken care of for you. The compiler knows the alignment of the types it's working with, thanks to the strong static type system.

     As you can see here, all POD types are trivial.

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   • What is the relaxation condition in graph theory

     Top Answer

     Answered Apr 07, 2010 · 43 votes

     Relaxation step:

     • You have two nodes, u and v
     • For every node, you have a tentative distance from the source node (for all nodes except for the source, it starts at positive infinity and it only decreases up to reaching its minimum).

     The relaxation step basically is asking this:

     • I already know that I can reach v with some path of distance dist[v]. Could I improve on this by going to v through u instead? (where the distance of the latter would be dist[u] + weight(u, v))

     Graphically:

     s ~~~~~~~> v \ ^ \ | \~~~~~> u-

     You know some path s~>v which has distance dist[v], and you know some path s~>u which has distance dist[u]. If dist[u] + weight(u, v) < dist[v], then the path s~>u->v is shorter than s~>v, so you'd better use that one!

     (I write a~>b to mean a path of any length from a to b, while a->b I mean a direct single edge from a to b).

     You may also want to check this lecture: http://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-046JFall-2005/VideoLectures/detail/embed17.htm

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   • What is a minimal path in a graph?

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     Answered Nov 15, 2008 · 6 votes

     Minimal path is the set of edges which when traversed cover the least amount of distance between two edges. Minimal distance is the sum of the distance between the edges of a minimal path.

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     4/5

   • What is a parity check matrix? (Information theory)

     Top Answer

     Answered May 18, 2010 · 3 votes

     I think your link explains it fairly well, but I'll try to simplify further.

     Let x be your message, a k-element row vector. Let G be your generator matrix, an k-by-n binary matrix where n > k. Let y be your n-element transmitted codeword where y = xG. Let z be your n-element received codeword.

     Hopefully, z = y. But when transmitting y across a noisy channel, it is possible for y to become corrupted, e.g., z != y.

     An (n-k)-by-n parity matrix H is applied to the received codeword z to check if z is valid. The vector w = zH' can detect up to a certain number of bit errors in z.

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3. www.geeksforgeeks.org · graph-types-and-applicationsTypes of Graphs with Examples - GeeksforGeeks

   www.geeksforgeeks.org · graph-types-and-applications

   

   • Finite Graphs. A graph is said to be finite if it has a finite number of vertices and a finite number of edges. A finite graph is a graph with a finite number of vertices and edges.
   • Infinite Graph: A graph is said to be infinite if it has an infinite number of vertices as well as an infinite number of edges.
   • Trivial Graph: A graph is said to be trivial if a finite graph contains only one vertex and no edge. A trivial graph is a graph with only one vertex and no edges.
   • Simple Graph: A simple graph is a graph that does not contain more than one edge between the pair of vertices. A simple railway track connecting different cities is an example of a simple graph.

4. People also ask

   What is a non trivial graph?

   • A non-trivial graph is any graph that contains more than one vertex. In other words, it consists of at least two distinct points (vertices) that may or may not be connected by edges. Non-trivial graphs exhibit a more complex structure with a higher degree of connectivity between their vertices compared to trivial graphs.

   Trivial Graph in Graph Theory

   www.prepbytes.com/blog/graphs/trivial-graph-in-graph-theory/

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   What is a non-trivial connected graph?

   • This graph meets the definition of connected vacuously (since an edge requires two vertices). A non-trivial connected graph is any connected graph that isn't this graph. A non-trivial connected component is a connected component that isn't the trivial graph, which is another way of say that it isn't an isolated point.

   terminology - What is a Non-Trivial Connected Graph? - Mathematics

   math.stackexchange.com/questions/2080962/what-is-a-non-trivial-connected-graph

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   How do you know if a graph is a trivial graph?

   • Degree of the Vertex: In graph theory, the degree of a vertex refers to the number of edges incident to that vertex. In the case of a trivial graph, the degree of the single vertex is 0 since there are no edges connected to it. 4. Connectivity: As there are no edges in a trivial graph, it is a disconnected graph.

   Trivial Graph in Graph Theory

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   What is a trivial edge in graph theory?

   • In graph theory, a trivial edge is an edge that connects a vertex to itself. Mathematically, it is represented as an edge between the same vertex (v, v) where ‘v’ is a vertex in the graph. Trivial edges do not add any new information or connectivity to the graph, as they create loops by connecting a vertex back to itself.

   Trivial Graph in Graph Theory

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5. math.stackexchange.com · questions · 1364996terminology - What is a nontrivial graph? - Mathematics Stack ...

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   Jul 17, 2015 · I use empty graph to mean a graph without edges, and therefore a nonempty graph would be a graph with at least one edge. According to both wikipedia.com and wikibooks.com, a trivial graph is a graph with 1 vertex and 0 edges.

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7. math.stackexchange.com · questions · 2080962terminology - What is a Non-Trivial Connected Graph ...

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   Jan 2, 2017 · A non-trivial connected graph is any connected graph that isn't this graph. A non-trivial connected component is a connected component that isn't the trivial graph, which is another way of say that it isn't an isolated point.

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9. www.cs.columbia.edu · ~cs4203 · filesOutline 1.1 Graphs and Digraphs 1.2 Common Families of Graphs ...

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   4 GRAPH THEORY { LECTURE 1 INTRODUCTION TO GRAPH MODELS Proposition 1.1. A non-trivial simple graph G must have at least one pair of vertices whose degrees are equal. Proof. pigeonhole principle Theorem 1.2 (Euler’s Degree-Sum Thm). The sum of the degrees of the vertices of a graph is twice the number of edges. Corollary 1.3.

10. math.mit.edu · research · highschoolGraph Theory and the Six Degrees of Separation - MIT Mathematics

    math.mit.edu · research · highschool

    

    In this paper, we will introduce the basics of graph theory and learn how it is applied to networks through the study of random graphs, which links the subjects of graph theory and probability together.

11. www.prepbytes.com · trivial-graph-in-graph-theoryTrivial Graph in Graph Theory

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    Aug 3, 2023 · Q1: What is a non-trivial graph in graph theory? A non-trivial graph is any graph that contains more than one vertex. In other words, it consists of at least two distinct points (vertices) that may or may not be connected by edges.

12. www.math.kit.edu · iag6 · lehreLecture Notes Graph Theory - KIT

    www.math.kit.edu · iag6 · lehre

    

    The graph Gis non-trivial if it contains at least one edge, i.e., E 6= ;. non-trivial Equivalently, Gis non-trivial if Gis not an empty graph. The order of G, denoted by jGj, is the number of vertices of G, i.e., jGj= jVj. order, jGj The size of G, denoted by kGk, is the number of edges of G, i.e., kGk= jEj. size, kGk