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1. Top results related to what is a non-trivial graph in science examples

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

     Top Answer

     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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     1/5

   • What is a minimal path in a graph?

     Top Answer

     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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     2/5

   • In a random graph: What is the probability that a node has a link to any node on a list x defined special nodes?

     Top Answer

     Answered Apr 05, 2016 · 4 votes

     For a non-special node, there are x potential edges from that node to a special node. For any such potential edge the probability of the edge not being in the graph is 1-p. Assuming independence, the probability that it avoids all special nodes is (1-p)^x. The complementary probability is what you seek, it is

     1 - (1-p)^x-

     For special nodes, the probability that a given special node is connected to one of the other special nodes is

     1 - (1-p)^(x-1)-

     You can combine these answers in various ways. Pick a node at random. The probability that it is either special or has an edge connecting it to a special node is:

     x/N + (N-x)/N * [1-(1-p)^x]-

     the probability that it has an edge connecting to a special node is:

     x/N * [1 - (1-p)^(x-1)] + (N-x)/N * [1 - (1-p)^x]-

     in all cases -- these tend to 1 as x tends to N.

     Since this is Stack Overflow, a bit of programming is in order. Here is a Python 3 Monte Carlo simulation that seems to suggest the accuracy of the formula for the probability that a randomly selected node is either special or adjacent to a special:

     import random-#The following function returns a random graph on nodes#0,1,...,N-1 where edges are chosen with probability p#The graph is returned as a dictionary keyed by the #The corresponding values are sorted lists of adjacent nodesdef randGraph(N,p): #initialize G: G = {} for i in range(N): G[i] = [] #iterate through potential edges: for i in range(N-1): for j in range(i+1,N): if random.random() < p: G[i].append(j) G[j].append(i) #sort adjacency lists before returning: for i in G: G[i].sort() return G#A function to determine the number of nodes#in a graph that are either#special or connected to a special,#where special means: 0,1,2,...,xdef specialsAndFriends(G,x): count = 0 for i in G: if (i<x) or (len(G[i]) > 0 and G[i][0] < x): count +=1 return count#now -- a Monte Carlo simulation:def estimateProb(N,p,x,trials = 1000): estimates = [] for i in range(trials): G = randGraph(N,p) estimates.append(specialsAndFriends(G,x)/N) return sum(estimates)/trials#compare with:def exactProb(N,p,x): return x/N + (N-x)/N * (1 - (1-p)**x)

     (Python 2 would need to adjust e.g. x/N to float(x)/N).

     Sample output:

     >>> estimateProb(100,0.25,10)0.9496800000000086>>> >>> exactProb(100,0.25,10)0.9493178367614746

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     3/5

   • 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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     4/5

   • What is the best way to represent a map or train station line as a graph in code?

     Top Answer

     Answered Feb 10, 2021 · 0 votes

     you need at least a class Node/Station, then you need either 1. a class Graph that contains a list of weighted connections between 2 Nodes, or 2. a list of nodes and each nodes has a list of weighted connections with node pointers.

     Usually you would want your graph API to be able to return the neighbors of a node* sorted by weights, so probably sort that by calling graph.build() after adding all the weights.

     There is no real gain in adding "coordinates" for the stations, unless its a real application, as its much easier to just set costs between stations than trying to come up with good positions for stations, otherwise just draw yourself a station map and label the edges yourself.

     I'm guessing you want something like graph.path(a, b), then with Dijkstra's algorithm you will easily be able to do that, I would recommend setting up the code to call the algo before coding too much, that way if you're about to represent the data in a bad way, you will know earlier.

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     5/5

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2. 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.

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

   math.stackexchange.com · questions · 1364996

   

   Jul 17, 2015 · 5. I've been operating happily under the definition that a nontrivial graph is a graph with at least two vertices for some time. Today I came upon a source which defined a nontrivial graph as a graph with one or more edges. Now I have all my ruffles in a bunch. What is the most widely accepted definition of a nontrivial graph?

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

   math.stackexchange.com · questions · 2080962

   

   Jan 2, 2017 · The trivial graph is the graph on one vertex. 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 ...

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

   math.mit.edu · research · highschool

   

   Similarly, any graph with an order >1 would be labeled non-trivial. De nition 6. Whenever two graphs are complimentary, G and G share a vertex set, and for every distinct pair of vertices (u, v) of G, uv is an edge of G but not G (see example below). De nition 7. A bipartite graph is a graph consisting of two disjoint sets where every vertex in one

9. testbook.com · maths · types-of-graphs-in-graph-theoryTypes of Graphs in Graph Theory: Subgraphs, Properties & Examples

   testbook.com · maths · types-of-graphs-in-graph-theory

   

   • Directed Graph in Graph Theory
   • Undirected Graph in Graph Theory
   • Null Graph in Graph Theory
   • Trivial Graph in Graph Theory
   • Simple Graph in Graph Theory
   • Complete Graph in Graph Theory
   • Connected Graph in Graph Theory
   • Disconnected Graph in Graph Theory
   • Regular Graph in Graph Theory
   • Cyclic Graph in Graph Theory

   A graph whose all edges are directed by arrows is known as a directed graph. They are also known as digraphs. In the above image, we can see a directed graph where all the edges are directed in a certain direction.

   See full list on testbook.com

   A graph whose edges are not directed by arrows is known as an undirected graph. In the above image, we see an undirected graph as its edges are not marked by arrows.

   See full list on testbook.com

   A graph in which there are no edges between its vertices is known as a null graph. It is also called an empty graph. A null graph with n number of vertices is denoted by NnNn. In the above image, three different null graphs are shown.

   See full list on testbook.com

   A graph with only one vertex is known as a trivial graph. In the above image, we see only one node and edges arising from it, thus it is a trivial graph.

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   A graph that is undirected and has no parallel edges orloopsis known as a simple graph. A simple graph with nvertices has the degree of every vertex is at most n-1. In the above image, we can see the difference between a simple and not a simple graph.

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   A graph in which every pair of vertices is joined by only one edge is called a completegraph. It contains all the possible edges. A complete graph with n number of vertices contains exactly nC2nC2 edges and is represented by KnKn. In the above image we see that each vertex in the graph is connected with all the remaining vertices through exactly on...

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   A connected graph is a graph where we can visit from any one vertex to any other vertex. In a connected graph, there is at least one edge or path that exists between every pair of vertices. In the above image we can traverse from any one vertex to any other vertex; it is a connected graph.

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   A disconnected graph is a graph in which no path exists between every pair of vertices. In the above image we see disconnected graphs.

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   A Regular graph is a graph in which the degree of all the vertices is the same. If the degree of all the vertices is k, then it is called a k-regular graph. In the above image all the vertices have degree 2 and thus it is a 2-regular graph.

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   A graph with n vertices and n edges forming a cycle of nwith all its edges is known as cycle graph. Any graph containing at least one cycle in it is known as a cyclic graph. In the above image the graph contains two cycles in it hence it is a cyclic graph.

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10. brilliant.org · wiki · graph-theoryGraph Theory | Brilliant Math & Science Wiki

    brilliant.org · wiki · graph-theory

    

    A non-trivial graph consists of one or more vertices (or nodes) connected by edges. Each edge connects exactly two vertices, although any given vertex need not be connected by an edge. The degree of a vertex is the number of edges connected to that vertex. In the graph below, vertex \( A \) is of degree 3, while vertices \( B \) and \( C \) are ...

11. People also ask

    What is a non-trivial graph?

    • A non-trivial graph consists of one or more vertices (or nodes) connected by edges. Each edge connects exactly two vertices, although any given vertex need not be connected by an edge. The degree of a vertex is the number of edges connected to that vertex. In the graph below, vertex A A is of degree 3, while vertices B B and C C are of degree 2.

    Graph Theory | Brilliant Math & Science Wiki

    brilliant.org/wiki/graph-theory/

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    What is a trivial graph?

    • 3. 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. It is also known as a singleton graph or a single vertex graph. A trivial graph is the simplest type of graph and is often used as a starting point for building more complex graphs.

    Types of Graphs with Examples - GeeksforGeeks

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    What is the difference between a trivial graph and a nonempty graph?

    • @ManuelLafond I agree. 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.

    What is a nontrivial graph? - Mathematics Stack Exchange

    math.stackexchange.com/questions/1364996/what-is-a-nontrivial-graph

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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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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