Meyer's law: Difference between revisions

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In [[graph theory]], the [[degree diameter problem]] is the problem of finding the largest possible [[graph (mathematics)|graph]] for a given maximum [[degree (graph theory)|degree]] and [[diameter (graph theory)|diameter]].  The [[Moore bound]] sets limits on this, but for many years mathematicians in the field have been interested in a more precise answer.  The table below gives current progress on this problem (excluding the case of degree 2, where the largest graphs are [[cycle (graph theory)|cycle]]s with an odd number of vertices).


==Table of the orders of the largest known graphs for the undirected degree diameter problem==
Below is the table of the vertex numbers for the best-known graphs (as of October 2008) in the undirected [[degree diameter problem]] for graphs of degree at most 3 ≤ ''d'' ≤ 16 and diameter 2 ≤ ''k'' ≤ 10. Only a few of the graphs in this table (marked in bold) are known to be optimal (that is, largest possible).  The remainder are merely the largest so far discovered, and thus finding a larger graph that is closer in order (in terms of the size of the vertex set) to the Moore bound is considered an [[open problem]]. Some general constructions are known for values of ''d'' and ''k'' outside the range shown in the table.
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''||  '''2''' ||  '''3''' ||  '''4'''|| '''5''' ||  '''6''' || '''7''' ||  '''8''' ||  '''9''' ||  '''10'''
|-
| '''3''' || style="background-color: red;" | '''10''' ||style="background-color: blue;" | '''20''' ||style="background-color: blue;" | '''38''' ||style="background-color: #66cc66;" | 70 ||style="background-color: #ffff00;" | 132 ||style="background-color: #ffff00;" | 196 ||style="background-color: #ffff00;" | 336 ||style="background-color: #ffff00;" | 600 ||style="background-color: #ffcc99;" | 1250
|-
| '''4''' ||style="background-color: blue;" | '''15''' ||style="background-color: #999900;" | 41 ||style="background-color: #ffff00;" | 96 || 364 ||style="background-color: #666666;" | 740 ||style="background-color: #FF9900;" | 1 320 ||style="background-color: #FF9900;" | 3 243 ||style="background-color: #FF9900;" | 7 575 ||style="background-color: #FF9900;" | 17 703
|-
| '''5''' ||style="background-color: blue;" | '''24''' ||style="background-color: #ffff00;" | 72 ||style="background-color: #bbffff;" | 210 ||style="background-color: #FF9900;" |  624 ||style="background-color: #00ff7f;" | 2 772 ||style="background-color: #FF9900;" |  5 516 ||style="background-color: #FF9900;" |  17 030 ||style="background-color: #FF9900;" |  57 840 ||style="background-color: #FF9900;" |  187 056
|-
| '''6''' ||style="background-color: #336666;" | '''32''' ||style="background-color: #ffff00;" | 110 ||style="background-color: #FF9900;" | 390 ||style="background-color: #FF9900;" | 1404 ||style="background-color: #00ff7f;" | 7 917 ||style="background-color: #FF9900;" | 19 383 ||style="background-color: #FF9900;" | 76 461 ||style="background-color: #FF9900;" | 307 845 ||style="background-color: #FF9900;" | 1 253 615
|-
| '''7''' ||style="background-color: red;" | '''50''' ||style="background-color: #ffff00;" | 168 ||style="background-color: #bbffff;" | 672 ||style="background-color: #FF0066;" | 2 756 ||style="background-color: #FF9900;" | 11 988 ||style="background-color: #FF9900;" | 52 768 ||style="background-color: #FF9900;" | 249 660 ||style="background-color: #FF9900;" | 1 223 050 ||style="background-color: #FF9900;" | 6 007 230
|-
| '''8''' || 57 ||style="background-color: #993300;" | 253 ||style="background-color: #FF9900;" | 1 100 ||style="background-color: #FF9900;" | 5 060 ||style="background-color: #666633;" | 39 672 ||style="background-color: #FF9900;" | 131 137 ||style="background-color: #FF9900;" | 734 820 ||style="background-color: #FF9900;" | 4 243 100 ||style="background-color: #FF9900;" | 24 897 161 
|-
| '''9''' || 74 || 585 ||style="background-color: #FF9900;" | 1 550 ||style="background-color: #FF9900;" | 8 200 ||style="background-color: #00ff7f;" | 75 893 ||style="background-color: #FF9900;" | 279 616 ||style="background-color: #FF9900;" | 1 686 600 ||style="background-color: #FF9900;" | 12 123 288 ||style="background-color: #FF9900;" | 65 866 350 
|-
| '''10''' || 91 || 650 ||style="background-color: #FF9900;" | 2 286 ||style="background-color: #FF9900;" | 13 140 ||style="background-color: #666633;" | 134 690 ||style="background-color: #FF9900;" | 583 083 ||style="background-color: #FF9900;" | 4 293 452 ||style="background-color: #FF9900;" | 27 997 191 ||style="background-color: #FF9900;" | 201 038 922
|-
| '''11''' ||style="background-color: #ffff00;" | 104 || 715 || 3 200 ||style="background-color: #FF9900;" | 19 500 || 156 864 ||style="background-color: #FF9900;" | 1 001 268 ||style="background-color: #FF9900;" | 7 442 328 || style="background-color: #FF9900;" | 72 933 102 ||style="background-color: #FF9900;" |  600 380 000
|-
| '''12''' || 133 ||style="background-color: #666633;" | 786 || 4 680 ||style="background-color: #FF9900;" | 29 470||style="background-color: #00ff7f;" | 359 772 ||style="background-color: #FF9900;" | 1 999 500 ||style="background-color: #FF9900;" | 15 924 326  ||style="background-color: #FF9900;" | 158 158 875 ||style="background-color: #FF9900;" | 1 506 252 500
|-
| '''13''' ||style="background-color: #ff99ff;" | 162 ||style="background-color: #666633;" | 851 || 6 560 ||style="background-color: #FF9900;" |40 260 || 531 440 ||style="background-color: #FF9900;" | 3 322 080 ||style="background-color: #FF9900;" | 29 927 790  ||style="background-color: #FF9900;" | 249 155 760 ||style="background-color: #FF9900;" | 3 077 200 700
|-
| '''14''' || 183 ||style="background-color: #666633;" | 916 || 8 200 ||style="background-color: #FF9900;" | 57 837 ||style="background-color: #00ff7f;" | 816 294 || 6 200 460 ||style="background-color: #FF9900;" | 55 913 932  ||style="background-color: #FF9900;" | 600 123 780 ||style="background-color: #FF9900;" | 7 041 746 081
|-
| '''15''' || 186 || 1 215 || 11 712 ||style="background-color: #FF9900;" | 76 518 || 1 417 248 ||style="background-color: #FF9900;" | 8 599 986 ||style="background-color: #FF9900;" | 90 001 236  ||style="background-color: #FF9900;" | 1 171 998 164 ||style="background-color: #FF9900;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #ffff00;" | 198 || 1 600  || 14 640 || 132 496 || 1 771 560 || 14 882 658 ||style="background-color: #FF9900;" | 140 559 416 ||style="background-color: #FF9900;" | 2 025 125 476 ||style="background-color: #FF9900;" | 12 951 451 931
|}
</center>


The following table is the key to the colors in the table presented above:
Have you see the My Pillow Pets comercial throughout Saturday morning cartoons? This adorable animal pilow that can turn into a huggable stuffed animal is soon to be all the rage. You could have even started seeing them in some stores around town, however get them now since they will certainly not be around for long and their cost will go up.<br><br>People might also make use of a lumber pillow while taking a trip. A U-shaped lumber pillow is light and portable. It supports the neck and head a person is seated and alleviates pressure on the muscles of the shoulders and neck. So, use it specifically, if an individual desires to oversleep a seated posture. The donut pillow has a hole in its center. This sort of pillow is for individuals who struggle with discomfort of the tailbone. Specifically, pregnant females establish tailbone discomfort. The donut pillow provides relief from pain. The central hole of the donut pillow soaks up the gravitational force and removes the pressure on the tailbone. turn your now pillow case the ideal side out through the back flaps, put a pillow form inside and enjoy your new luscious ruffles!<br><br>Post body (HTML version) There are a lot of facets of selecting a mattress that purchasing a brand-new one can end up being a complicated headache. Bed mattress have numerous qualities that each customer need to think about while trying to find the perfect match for the person. Firm or soft, memory foam or conventional coil-spring, and also luxurious, pillow-top, or euro-top. The language of the world of bed mattress can be puzzling. Many consumers may believe that plush, pillow-top, and euro-top are all really the exact same characteristic. Not so, and choosing in between the three is necessary to finding the best mattress to get quality sleep.<br><br>For allergy sufferers, utilizing the inaccurate pillow can trigger numerous sleepless nights packed with red, drippy eyes and nasal congestion. A variety of manufacturers have actually developed allergy-friendly pillows. These pillows do not bring in dust mites, which are the leading reason for all home allergies. Specialists advise different pillows for allergy victims, depending upon exactly what the allergy is. Home Dust and Dust Mites Jewel Sparkles puts on fabrics made from the scraps of an old Princess outfit. She has an animal Persian cat and loves to flaunt her sparkly clothing and boss individuals around. Her house is a rather pink material residence inlaid with diamonds. Her driveway is lined with giant pearls, and she makes use of a velvet cushion as a step up to her door.<br><br>The term "orthopedic" is synonymous to therapeutic. This indicates that the pillows are made not just to relieve you from body pain. It is also made to ease you from anxiety from the day's activities. Its healing abilities bring you to deep sleep. Every My Pillow Pet is a plush stuffed animal that doubles as a cushion. There is an accessory strap under the Animal's stomach that, when connected, really makes the Animal appear like it has legs, making it a treasured buddy. Merely unfasten that adapter and you instantly alter the animal into a pillowy, soft cushion that you can now lie your head on! Put the ribbon in the middle of the pillow cover, slowly gluing the ribbon in location to produce a swirling pattern. Quilted Feather Bed (Or Pillow Top)<br><br>Your wedding event will be one of the most unforgettable and unique moment of your life and ensuring that it is best is critical to all. If you would like to amaze your visitors with a little thank you present utilizing a pillow box to package the product could be the perfect options. They can fit a range of products inside of them making it much easier and quicker to package the gift. Little surprise table gifts at your wedding could just include those little touches that imply the world to your visitors, pillow boxes are appropriate for chocolates, thank you notes and many other little gift items.<br><br>The most common cause of hip pain is osteoarthritis, which results from wear and tear on the hip joint. Problems in the lower back, such as lumbar muscle spasm, can trigger hip discomfort since the two locations share nerves that can become inflamed. A variety of methods be followed to lower or alleviate hip discomfort. Good body mechanics are important to prevent reinjury. Avoid activities that need you to spread out the hips widely. Avoid lengthened sitting and restriction duplicated flexing at the hip. A Pillow for the Back<br><br>Amongst the different goose down pillows, this down pillow from 'Pike Street' is thought to be the best. It has medium density and features an extremely lovely damask Swiss dot patterned cover. It has a cotton cover, which is double sewn and has a silk piping surface. The medium density of the pillow provides great support for the head and neck, and helps in keeping correct spinal column alignment while resting. A lot of down pillow testimonials speak about the thread count. However, there is barely any relationship in between a higher thread count and fewer feathers getting away. Many pillow business offer a 1 month refund policy. Therefore, you will certainly have the ability to return the pillow in case you are dissatisfied with the item. Mattress Ticking and Quilting<br><br>Stuff your pillow in to the pillow case and enjoy the terrific treasure you have just developed. I hope you enjoy this crochet craft project and if you are looking for even more craft and hobby concepts, make sure to click here A Breastfeeding Pillow helps the nursing mom to take pleasure in the procedure of nursing her infant and to avoid complications such as back pain. Likewise Pregnancy Clothes as nursing tops and breastfeeding covers benefit mommies. If someone chose pillows were being taken for given and he gathered all the pillows in the world, envision. Can you think of how uneasy it would be? We've been spoiled for numerous years having that nice fluffy object under our head. The best ways to Stitch Flanges on a Pillow Case Good Beds for Side Sleepers<br><br>Pillows offer different levels of firmness. With a Down pillow you can get one that is extremely thin or really thick. Which you would select is based on your choice and how you sleep. Memory Foam Pillows are available in various shapes, firmnesses, and densities. For a pillow, 3 lb. density seems to work best. It is soft, yet supportive. Remember that we are discussing pillows and not a bed mattress. Exactly what is good enough to support your head is not always finest for your whole body. A shape pillow can be organized to suit you no matter how you sleep.
 
<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: red; text-align: center;" | * || The [[Petersen graph|Petersen]] and [[Hoffman–Singleton graph|Hoffman–Singleton]] graphs.
|-
|style="background-color: blue; text-align: center;" | * || Optimal graphs which are not [[Moore graph]]s.  
|-
|style="background-color: #999900; text-align: center;" | * || Graph found by James Allwright.
|-
|style="background-color: #336666; text-align: center;" | * || Graph found by G. Wegner.  
|-
|style="background-color: #ffff00; text-align: center;" | * || Graphs found by Geoffrey Exoo.
|-
|style="background-color: #ff99ff; text-align: center;" | * || Family of graphs found by Brendan D. McKay, Mirka Miller and Jozef Širáň. More details are available in a paper by the authors.
|-
|style="background-color: #666633; text-align: center;" | * || Graphs found by J. Gómez.  
|-
|style="background-color: #993300; text-align: center;" | * || Graph found by Mitjana M. and Francesc Comellas. This graph was also found independently by Michael Sampels.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graph found by Fiol, M.A. and Yebra, J.L.A.
|-
|style="background-color: #666666; text-align: center;" | * || Graph found by Francesc Comellas and J. Gómez.
|-
|style="background-color: #00ff7f; text-align: center;" | * || Graphs found by G. Pineda-Villavicencio, J. Gómez, Mirka Miller and H. Pérez-Rosés. More details are available in a paper by the authors.
|-
|style="background-color: #FF9900; text-align: center;" | * || Graphs found by Eyal Loz. More details are available in a paper by Eyal Loz and Jozef Širáň.  
|-
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by Michael Sampels.
|-
|style="background-color: #FF0066; text-align: center;" | * || Graphs found by Michael J. Dinneen and Paul Hafner. More details are available in a paper by the authors.
|-
|style="background-color: #ffcc99; text-align: center;" | * || Graph found by [[Marston Conder]].
|}
</center>
 
==References==
* {{citation
  | last1 = Hoffman
  | first1 = Alan J.
  | last2 = Singleton
  | first2 = Robert R.
  | title = Moore graphs with diameter 2 and 3
  | journal = IBM Journal of Research and Development
  | volume = 5
  | issue = 4
  | year = 1960
  | pages = 497–504
  | url = http://www.research.ibm.com/journal/rd/045/ibmrd0405H.pdf
  | mr = 0140437
  | doi=10.1147/rd.45.0497}}
 
* {{citation
  | last1 = J. Dinneen
  | first1 = Michael
  | last2 = Hafner
  | first2 = Paul R.
  | title = New Results for the Degree/Diameter Problem
  | journal = Networks
  | volume = 24
  | issue = 7
  | year = 1994
  | pages = 359&ndash;367
  | arxiv = math/9504214
  | doi = 10.1002/net.3230240702}}
 
* {{citation
  | last1 = McKay
  | first1 = Brendan D.
  | last2 = Miller
  | first2 = Mirka
  | last3 = Širáň
  | first3 = Jozef 
  | title = A note on large graphs of diameter two and given maximum degree
  | journal = Journal of Combinatorial Theory Series B
  | volume = 74
  | issue = 4
  | year = 1998
  | pages = 110&ndash;118
  | url = http://portal.acm.org/citation.cfm?id=299331
  | doi = 10.1006/jctb.1998.1828}}
 
* {{citation
  | last1 = Miller
  | first1 = Mirka
  | last2 = Širáň
  | first2 = Jozef 
  | title =  Moore graphs and beyond: A survey of the degree/diameter problem
  | journal = Electronic Journal of Combinatorics
  | volume = Dynamic survey D
  | year = 2005
  | url = http://www.combinatorics.org/Surveys/ds14.pdf}}
 
* {{citation
  | last1 = Pineda-Villavicencio
  | first1 = Guillermo
  | last2 = Gómez
  | first2 = José
  | last3 = Miller
  | first3 = Mirka
  | last4 = Pérez-Rosésd
  | first4 = Hebert
  | title = New Largest Graphs of Diameter 6
  | journal = Electronic Notes in Discrete Mathematics
  | volume = 24
  | year = 2006
  | pages = 153&ndash;160
  | doi = 10.1016/j.endm.2006.06.044}}
 
* {{citation
  | last1 = Loz
  | first1 = Eyal
  | last2 = Širáň
  | first2 = Jozef 
  | title = New record graphs in the degree-diameter problem
  | journal = Australasian Journal of Combinatorics
  | volume = 41
  | year = 2008
  | pages = 63&ndash;80
  | url = http://ajc.maths.uq.edu.au/pdf/41/ajc_v41_p063.pdf
}}
 
==External links==
* [http://www-mat.upc.es/grup_de_grafs/ Degree Diameter] online table.
* [http://combinatoricswiki.org/wiki/The_Degree/Diameter_Problem  The Degree - Diameter Problem on CombinatoricsWiki.org].
* [http://www.eyal.com.au/wiki/The_Degree/Diameter_Problem Eyal Loz's] Degree-Diameter problem page.
* [http://isu.indstate.edu/ge/DD/index.html Geoffrey Exoo's] Degree-Diameter record graphs page.
* [http://guillermo.com.au/wiki/Research Guillermo Pineda-Villavicencio's] Research page.
 
[[Category:Graph theory]]
[[Category:Top lists]]

Latest revision as of 03:10, 5 June 2014


Have you see the My Pillow Pets comercial throughout Saturday morning cartoons? This adorable animal pilow that can turn into a huggable stuffed animal is soon to be all the rage. You could have even started seeing them in some stores around town, however get them now since they will certainly not be around for long and their cost will go up.

People might also make use of a lumber pillow while taking a trip. A U-shaped lumber pillow is light and portable. It supports the neck and head a person is seated and alleviates pressure on the muscles of the shoulders and neck. So, use it specifically, if an individual desires to oversleep a seated posture. The donut pillow has a hole in its center. This sort of pillow is for individuals who struggle with discomfort of the tailbone. Specifically, pregnant females establish tailbone discomfort. The donut pillow provides relief from pain. The central hole of the donut pillow soaks up the gravitational force and removes the pressure on the tailbone. turn your now pillow case the ideal side out through the back flaps, put a pillow form inside and enjoy your new luscious ruffles!

Post body (HTML version) There are a lot of facets of selecting a mattress that purchasing a brand-new one can end up being a complicated headache. Bed mattress have numerous qualities that each customer need to think about while trying to find the perfect match for the person. Firm or soft, memory foam or conventional coil-spring, and also luxurious, pillow-top, or euro-top. The language of the world of bed mattress can be puzzling. Many consumers may believe that plush, pillow-top, and euro-top are all really the exact same characteristic. Not so, and choosing in between the three is necessary to finding the best mattress to get quality sleep.

For allergy sufferers, utilizing the inaccurate pillow can trigger numerous sleepless nights packed with red, drippy eyes and nasal congestion. A variety of manufacturers have actually developed allergy-friendly pillows. These pillows do not bring in dust mites, which are the leading reason for all home allergies. Specialists advise different pillows for allergy victims, depending upon exactly what the allergy is. Home Dust and Dust Mites Jewel Sparkles puts on fabrics made from the scraps of an old Princess outfit. She has an animal Persian cat and loves to flaunt her sparkly clothing and boss individuals around. Her house is a rather pink material residence inlaid with diamonds. Her driveway is lined with giant pearls, and she makes use of a velvet cushion as a step up to her door.

The term "orthopedic" is synonymous to therapeutic. This indicates that the pillows are made not just to relieve you from body pain. It is also made to ease you from anxiety from the day's activities. Its healing abilities bring you to deep sleep. Every My Pillow Pet is a plush stuffed animal that doubles as a cushion. There is an accessory strap under the Animal's stomach that, when connected, really makes the Animal appear like it has legs, making it a treasured buddy. Merely unfasten that adapter and you instantly alter the animal into a pillowy, soft cushion that you can now lie your head on! Put the ribbon in the middle of the pillow cover, slowly gluing the ribbon in location to produce a swirling pattern. Quilted Feather Bed (Or Pillow Top)

Your wedding event will be one of the most unforgettable and unique moment of your life and ensuring that it is best is critical to all. If you would like to amaze your visitors with a little thank you present utilizing a pillow box to package the product could be the perfect options. They can fit a range of products inside of them making it much easier and quicker to package the gift. Little surprise table gifts at your wedding could just include those little touches that imply the world to your visitors, pillow boxes are appropriate for chocolates, thank you notes and many other little gift items.

The most common cause of hip pain is osteoarthritis, which results from wear and tear on the hip joint. Problems in the lower back, such as lumbar muscle spasm, can trigger hip discomfort since the two locations share nerves that can become inflamed. A variety of methods be followed to lower or alleviate hip discomfort. Good body mechanics are important to prevent reinjury. Avoid activities that need you to spread out the hips widely. Avoid lengthened sitting and restriction duplicated flexing at the hip. A Pillow for the Back

Amongst the different goose down pillows, this down pillow from 'Pike Street' is thought to be the best. It has medium density and features an extremely lovely damask Swiss dot patterned cover. It has a cotton cover, which is double sewn and has a silk piping surface. The medium density of the pillow provides great support for the head and neck, and helps in keeping correct spinal column alignment while resting. A lot of down pillow testimonials speak about the thread count. However, there is barely any relationship in between a higher thread count and fewer feathers getting away. Many pillow business offer a 1 month refund policy. Therefore, you will certainly have the ability to return the pillow in case you are dissatisfied with the item. Mattress Ticking and Quilting

Stuff your pillow in to the pillow case and enjoy the terrific treasure you have just developed. I hope you enjoy this crochet craft project and if you are looking for even more craft and hobby concepts, make sure to click here A Breastfeeding Pillow helps the nursing mom to take pleasure in the procedure of nursing her infant and to avoid complications such as back pain. Likewise Pregnancy Clothes as nursing tops and breastfeeding covers benefit mommies. If someone chose pillows were being taken for given and he gathered all the pillows in the world, envision. Can you think of how uneasy it would be? We've been spoiled for numerous years having that nice fluffy object under our head. The best ways to Stitch Flanges on a Pillow Case Good Beds for Side Sleepers

Pillows offer different levels of firmness. With a Down pillow you can get one that is extremely thin or really thick. Which you would select is based on your choice and how you sleep. Memory Foam Pillows are available in various shapes, firmnesses, and densities. For a pillow, 3 lb. density seems to work best. It is soft, yet supportive. Remember that we are discussing pillows and not a bed mattress. Exactly what is good enough to support your head is not always finest for your whole body. A shape pillow can be organized to suit you no matter how you sleep.