Editing Higman–Sims graph (section)

==Algebraic properties==
The [[automorphism group]] of the Higman–Sims graph is a group of order {{formatnum:88704000}} isomorphic to the [[semidirect product]] of the [[Higman–Sims group]] of order {{formatnum:44352000}} with the [[cyclic group]] of order 2.<ref>{{cite web
  | url = http://www.win.tue.nl/~aeb/drg/graphs/Higman-Sims.html
  | title = Higman&ndash;Sims graph
  | author-link = Andries E. Brouwer
  | author = Brouwer, Andries E.}}</ref> It has automorphisms that take any edge to any other edge, making the Higman–Sims graph an [[edge-transitive graph]].<ref>Brouwer, A. E. and Haemers, W. H. "The Gewirtz Graph: An Exercise in the Theory of Graph Spectra." Euro. J. Combin. 14, 397&ndash;407, 1993.</ref> The outer elements induce odd permutations on the graph. As mentioned [[#From Hoffman–Singleton graph|above]], there are 352 ways to partition the Higman–Sims graph into a pair of Hoffman–Singleton graphs; these partitions actually come in 2 orbits of size 176 each, and the outer elements of the Higman–Sims group swap these orbits.<ref>{{Cite book| title = Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups
 | last1 = Conway | first1 = J. H.
 | last2 = Curtis | first2 = R. T.
 | last3 = Norton | first3 = S. P.
 | last4 = Parker | first4 = R. A.
 | last5 = Wilson | first5 = R. A.
 | author1-link = John Horton Conway
 | author3-link = Simon P. Norton
 | author4-link = Richard A. Parker
 | author5-link = Robert Arnott Wilson
 | year = 1985
 | others = with computational assistance from J. G. Thackray
 | publisher = Oxford University Press
 | isbn = 978-019853199-9
 | ref = none
}}</ref>

The characteristic polynomial of the Higman–Sims graph is (''x''&nbsp;&minus;&nbsp;22)(''x''&nbsp;&minus;&nbsp;2)<sup>77</sup>(''x''&nbsp;+&nbsp;8)<sup>22</sup>. Therefore, the Higman–Sims graph is an [[integral graph]]: its [[Spectral graph theory|spectrum]] consists entirely of integers. It is also the only graph with this characteristic polynomial, making it a graph determined by its spectrum.