Is it true that central vertices (adjacent central vertices) of a graph always lie in every diametral path? Few graphs satisfy this property. But no idea how to prove it? Can anyone provide some hint/help to prove/disprove it?
Central vertices lie in the diametral path if they ar adjacent in the graph?????
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combinatorics
discrete-mathematics
graph-theory
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1Not every central vertex lie on every diametral path. Consider $C_4$. – 2017-02-09
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0Oh yes.... thank you very much. – 2017-02-09
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0@Smylic edited my question.... can you help now please. – 2017-02-09
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1More strong statement: it is false that every central vertex lie on at least one central path. Consider $G = (\{\,1, 2, 3, 4, 5\,\}, \{\,\{\,1, 2\,\}, \{\,2, 3\,\}, \{\,3, 4\,\}, \{\,2, 5\,\}, \{\,3, 5\,\}\,\})$. Also one pair of adjacent central vertices belongs to the only diametal path, but two other pairs doesn't. – 2017-02-09
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0Thanks for the clarification. – 2017-02-09
1 Answers
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The claim "central vertices of a graph always lie in every diametral path" is not entirely clear, but I can't come up with an interpretation that is true.
If you mean
If $v$ is a central vertex, then every diametral path contains $v$.
then that is certainly not true in general. There are plenty of counterexamples where every vertex is central but no vertex is on every diameter: Cycle graphs, complete graphs, the graph of every regular polyhedron ...
It is not even true that
Every diametral path contains at least one central vertex.
which has this counterexample:
A---B
|\ /|
| C |
|/ \|
D---E
where C is the only central vertex, but the path A-B-E is a diameter.
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0...Thanks a lot for answer. It is clear to me now. – 2017-02-09