Editing Higman–Sims graph (section)

{{CS1 config|mode=cs1}}
{{infobox graph
 | name = Higman–Sims graph
 | image = [[Image:Higman Sims Graph.svg|220px]]
 | image_caption = Drawing based on Paul R. Hafner's construction.<ref>{{Cite journal
 | last1 = Hafner | first1 = P. R.
 | title = On the Graphs of Hoffman&ndash;Singleton and Higman&ndash;Sims
 | journal = The Electronic Journal of Combinatorics
 | volume = 11 | issue = 1 | year = 2004 | pages = R77(1–32)
 | doi = 10.37236/1830
 | url=http://www.combinatorics.org/Volume_11/PDF/v11i1r77.pdf
 | doi-access = free
 }}.</ref>
 | namesake = [[Donald G. Higman]]<br/>[[Charles C. Sims]]
 | vertices = 100
 | edges = 1100
 | radius = 2
 | diameter = 2
 | girth = 4
 | automorphisms = {{formatnum:88704000}} ([[Higman–Sims group|HS]]:2)
 | chromatic_number = 
 | chromatic_index = 
 | properties = [[Strongly regular graph|Strongly regular]]<br>[[Edge-transitive graph|Edge-transitive]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[Eulerian graph|Eulerian]]
}}

[[Image:Higman Sims Graph Parts.svg|220px|right|thumb|
The separated parts of Hafner's construction.]]

In mathematical [[graph theory]], the '''Higman–Sims graph''' is a 22-[[regular graph|regular]] [[undirected graph]] with 100 vertices and 1100 edges. It is the unique [[strongly regular graph]] srg(100,22,0,6), where no neighboring pair of vertices share a common neighbor and each non-neighboring pair of vertices share six common neighbors.<ref>{{MathWorld|urlname=Higman-SimsGraph|title=Higman&ndash;Sims Graph}}</ref> It was first constructed by {{harvtxt|Mesner|1956}}<ref>{{cite thesis | last1=Mesner | first1=Dale Marsh | title=An investigation of certain combinatorial properties of partially balanced incomplete block experimental designs and association schemes, with a detailed study of designs of Latin square and related types | publisher=Department of Statistics, Michigan State University | type=Doctoral Thesis |mr=2612633 | year=1956}}
</ref> and rediscovered in 1968 by Donald G. Higman and Charles C. Sims as a way to define the [[Higman–Sims group]], a subgroup of [[Index of a subgroup|index]] two in the group of automorphisms of the Hoffman–Singleton graph.<ref>{{Cite journal
 | last1 = Higman | first1 = Donald G.
 | last2 = Sims | first2 = Charles C. | author2-link = Charles Sims (mathematician)
 | title = A simple group of order 44,352,000
 | journal = [[Mathematische Zeitschrift]]
 | volume = 105 | issue = 2 | year = 1968 | pages = 110–113
 | doi = 10.1007/BF01110435
 | hdl = 2027.42/46258
 | s2cid = 32803979
 | url = https://deepblue.lib.umich.edu/bitstream/2027.42/46258/1/209_2005_Article_BF01110435.pdf| hdl-access = free}}.</ref>