In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, the Hall–Janko graph, also known as the Hall-Janko-Wales graph, is a 36-
regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
undirected graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...
with 100 vertices and 1800 edges.
It is a
rank 3 strongly regular graph
In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that:
* Every two adjacent vertices have comm ...
with parameters (100,36,14,12) and a maximum
coclique of size 10. This parameter set is not unique, it is however uniquely determined by its parameters as a rank 3 graph. The Hall–Janko graph was originally constructed by D. Wales to establish the existence of the
Hall-Janko group as an
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
2 subgroup of its
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is th ...
.
The Hall–Janko graph can be constructed out of objects in U
3(3), the simple group of order 6048:
* In U
3(3) there are 36 simple maximal subgroups of order 168. These are the vertices of a subgraph, the U
3(3) graph. A 168-subgroup has 14 maximal subgroups of order 24, isomorphic to S
4. Two 168-subgroups are called adjacent when they intersect in a 24-subgroup. The U
3(3) graph is strongly regular, with parameters (36,14,4,6)
* There are 63 involutions (elements of order 2). A 168-subgroup contains 21 involutions, which are defined to be neighbors.
* Outside U
3(3) let there be a 100th vertex C, whose neighbors are the 36 168-subgroups. A 168-subgroup then has 14 common neighbors with C and in all 1+14+21 neighbors.
* An involution is found in 12 of the 168-subgroups. C and an involution are non-adjacent, with 12 common neighbors.
* Two involutions are defined as adjacent when they generate a dihedral subgroup of order 8.
[Robert A. Wilson, 'The Finite Simple Groups', Springer-Verlag (2009), p. 224.] An involution has 24 involutions as neighbors.
The characteristic polynomial of the Hall–Janko graph is
. Therefore the Hall–Janko graph is an
integral graph In the mathematical field of graph theory, an integral graph is a graph whose adjacency matrix's spectrum consists entirely of integers. In other words, a graph is an integral graph if all of the roots of the characteristic polynomial of its adjace ...
: its
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
consists entirely of integers.
References
{{DEFAULTSORT:Hall-Janko Graph
Group theory
Individual graphs
Regular graphs