Graphs with $operatorname{diam}(G)=2operatorname{rad}(G)$












1












$begingroup$


When is the diameter of a graph equal to twice of radius? I am currently studying graph theory and have faced many questions related to graphs with the mentioned property. Is there any general class of graphs which follow this property?



I know path graphs with odd vertices are such graphs, but is there a more general graph?






graph-theory







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
    $endgroup$
    – munchhausen
    Nov 13 '18 at 18:00






  • 1




    $begingroup$
    Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
    $endgroup$
    – aleph_two
    Dec 23 '18 at 5:38








  • 1




    $begingroup$
    (e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
    $endgroup$
    – aleph_two
    Dec 23 '18 at 6:30


















1












$begingroup$


When is the diameter of a graph equal to twice of radius? I am currently studying graph theory and have faced many questions related to graphs with the mentioned property. Is there any general class of graphs which follow this property?



I know path graphs with odd vertices are such graphs, but is there a more general graph?






graph-theory







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
    $endgroup$
    – munchhausen
    Nov 13 '18 at 18:00






  • 1




    $begingroup$
    Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
    $endgroup$
    – aleph_two
    Dec 23 '18 at 5:38








  • 1




    $begingroup$
    (e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
    $endgroup$
    – aleph_two
    Dec 23 '18 at 6:30
















1












1








1





$begingroup$


When is the diameter of a graph equal to twice of radius? I am currently studying graph theory and have faced many questions related to graphs with the mentioned property. Is there any general class of graphs which follow this property?



I know path graphs with odd vertices are such graphs, but is there a more general graph?






graph-theory







share|cite|improve this question











$endgroup$




When is the diameter of a graph equal to twice of radius? I am currently studying graph theory and have faced many questions related to graphs with the mentioned property. Is there any general class of graphs which follow this property?



I know path graphs with odd vertices are such graphs, but is there a more general graph?






graph-theory




graph-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 13 '18 at 10:40









Bernard

124k742117




124k742117










asked Nov 13 '18 at 9:58









Ankit KumarAnkit Kumar

1,542221




1,542221








  • 1




    $begingroup$
    A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
    $endgroup$
    – munchhausen
    Nov 13 '18 at 18:00






  • 1




    $begingroup$
    Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
    $endgroup$
    – aleph_two
    Dec 23 '18 at 5:38








  • 1




    $begingroup$
    (e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
    $endgroup$
    – aleph_two
    Dec 23 '18 at 6:30
















  • 1




    $begingroup$
    A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
    $endgroup$
    – munchhausen
    Nov 13 '18 at 18:00






  • 1




    $begingroup$
    Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
    $endgroup$
    – aleph_two
    Dec 23 '18 at 5:38








  • 1




    $begingroup$
    (e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
    $endgroup$
    – aleph_two
    Dec 23 '18 at 6:30










1




1




$begingroup$
A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
$endgroup$
– munchhausen
Nov 13 '18 at 18:00




$begingroup$
A tree has this property if and only if it has a center vertex. By a center vertex, I mean, consider iteratively removing leaves, i.e. at step one, remove all leaves, then at step 2 repeat. In a tree, you'll either be left with an edge or with a vertex. If you're left with a vertex, then that's called the center vertex of the tree. You can quickly check that a center vertex of a tree is exactly the realizer that $diam(G)=2rad(G)$.
$endgroup$
– munchhausen
Nov 13 '18 at 18:00




1




1




$begingroup$
Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
$endgroup$
– aleph_two
Dec 23 '18 at 5:38






$begingroup$
Generally these sorts of extremal questions can get pretty thorny and so are considered pretty specialized; you might get more attention from experts if you ask at MathOverflow.
$endgroup$
– aleph_two
Dec 23 '18 at 5:38






1




1




$begingroup$
(e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
$endgroup$
– aleph_two
Dec 23 '18 at 6:30






$begingroup$
(e.g. this property was shown for interval graphs with even diameter— fairly recently, and with quite a bit of effort.)
$endgroup$
– aleph_two
Dec 23 '18 at 6:30












1 Answer
1






active

oldest

votes


















1












$begingroup$

As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:




  • A tree has this property if and only if it has a center vertex.

  • An interval graph has this property if and only if its diameter is even.


The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks for the paper by the way. It's nice!
    $endgroup$
    – Ankit Kumar
    Dec 23 '18 at 6:39












Your Answer








StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996545%2fgraphs-with-operatornamediamg-2-operatornameradg%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:




  • A tree has this property if and only if it has a center vertex.

  • An interval graph has this property if and only if its diameter is even.


The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks for the paper by the way. It's nice!
    $endgroup$
    – Ankit Kumar
    Dec 23 '18 at 6:39
















1












$begingroup$

As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:




  • A tree has this property if and only if it has a center vertex.

  • An interval graph has this property if and only if its diameter is even.


The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks for the paper by the way. It's nice!
    $endgroup$
    – Ankit Kumar
    Dec 23 '18 at 6:39














1












1








1





$begingroup$

As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:




  • A tree has this property if and only if it has a center vertex.

  • An interval graph has this property if and only if its diameter is even.


The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.






share|cite|improve this answer











$endgroup$



As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:




  • A tree has this property if and only if it has a center vertex.

  • An interval graph has this property if and only if its diameter is even.


The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 23 '18 at 6:42

























answered Dec 23 '18 at 6:37









aleph_twoaleph_two

27412




27412












  • $begingroup$
    Thanks for the paper by the way. It's nice!
    $endgroup$
    – Ankit Kumar
    Dec 23 '18 at 6:39


















  • $begingroup$
    Thanks for the paper by the way. It's nice!
    $endgroup$
    – Ankit Kumar
    Dec 23 '18 at 6:39
















$begingroup$
Thanks for the paper by the way. It's nice!
$endgroup$
– Ankit Kumar
Dec 23 '18 at 6:39




$begingroup$
Thanks for the paper by the way. It's nice!
$endgroup$
– Ankit Kumar
Dec 23 '18 at 6:39


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996545%2fgraphs-with-operatornamediamg-2-operatornameradg%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Plaza Victoria

Image
Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l...
Read more

Puebla de Zaragoza

Image
Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
Read more

Musa

Image
Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
Read more