What Is A Simple Graph

What Is A Simple Graph Every complete graph is also a simple graph However between any two distinct vertices of a complete graph there is always exactly one edge between any two distinct vertices of a simple graph there is always at most one edge

Otherwise I d interpret simple with no loops as being equivalent to a tree But in any case the OP didn t specify simple or no loops So it would seem the answer to the question is it s not provable because it s not true endgroup A connected graph is a graph with no disjoint subgraphs A simple graph is a graph with no loops or multiple edges Easy Question What are the necessary and sufficient conditions on the order number of vertices and size number of edges of a given graph in order to ensure that it is simple and connected

What Is A Simple Graph

What Is A Simple Graph
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In graph theory what is the difference between a cycle and a simple cycle My impression is that a simple cycle is the same as a cycle except that you cannot repeat vertices As Omnomnomnom posted there are only 11 One way to approach this solution is to break it down by the number of edges on each graph A simple graph on 4 vertices can have at most 4 choose 2 6 edges 0 edges 1 unique graph 1 edge 1 unique graph

Dirac s theorem on Hamiltonian cycles the statement that an n vertex graph in which each vertex has degree at least n 2 must have a Hamiltonian cycle So a cycle exists that passes through all the vertices If you have a self complementary graph of order 4n half the vertices must each lie on fewer than 2n edges and the other half must lie on 2n or more edges Add a vertex by connecting it to the 2n vertices lying on fewer than 2n edges The result is a self complementary graph of order 4n 1

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Let G be a graph with n vertices Assume G is not connected so there are at least two connecting components Denote the number of vertices of the components with u und v begingroup Note I assumed the graph is connected in case a disconnected graph would be harder to think about but you can extend this logic to a planar graph by still assuming each vertex has degree greater than 5 and understanding that each component of a disconnected graph is connected endgroup

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Every complete graph is also a simple graph However between any two distinct vertices of a complete graph there is always exactly one edge between any two distinct vertices of a simple graph there is always at most one edge

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Otherwise I d interpret simple with no loops as being equivalent to a tree But in any case the OP didn t specify simple or no loops So it would seem the answer to the question is it s not provable because it s not true endgroup

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What Is A Simple Graph - As Omnomnomnom posted there are only 11 One way to approach this solution is to break it down by the number of edges on each graph A simple graph on 4 vertices can have at most 4 choose 2 6 edges 0 edges 1 unique graph 1 edge 1 unique graph