Graph details

Adjacency matrix
011110

101101

110011

110011

101101

011110


Adjacency list
1:

2
3
4
5

2:

1
3
4
6

3:

1
2
5
6

4:

1
2
5
6

5:

1
3
4
6

6:

2
3
4
5


HoG graph id
226
Graph name
Octahedral Graph
Graph submitted by
GraPHedron
Invariant values
The definitions of the invariants can be found
here.
Invariant 
Value 
Invariant 
Value 
Acyclic

No

Index

4

Algebraic Connectivity

4

Laplacian Largest Eigenvalue

6

Average Degree

4

Longest Induced Cycle

4

Bipartite

No

Longest Induced Path

2

Chromatic Index

4

Matching Number

3

Chromatic Number

3

Maximum Degree

4

Circumference

6

Minimum Degree

4

ClawFree

Yes

Minimum Dominating Set

2

Clique Number

3

Number of Components

1

Connected

Yes

Number of Edges

12

Density

0.8

Number of Triangles

8

Diameter

2

Number of Vertices

6

Edge Connectivity

4

Planar

Yes

Eulerian

Yes

Radius

2

Genus

0

Regular

Yes

Girth

3

Second Largest Eigenvalue

0

Hamiltonian

Yes

Smallest Eigenvalue

2

Independence Number

2

Vertex Connectivity

4

A table row rendered like this
indicates that the graph is marked as being interesting for that invariant.
Comments
Posted by GraPHedron at Mar 28, 2012 11:56 AM.
Is an extremal graph found by GraPHedron. See "H. Melot, Facet defining inequalities among graph invariants: the system graphedron. Discrete Applied Mathematics 156 (2008), 18751891" for more information.
Posted by MathWorld at Mar 28, 2012 11:58 AM.
Source: MathWorld and GraphData in Mathematica.
Posted by Gunnar Brinkmann at May 15, 2014 3:11 PM.
keyword: min_hc_triang_sep
The unique triangulation with 6 vertices and no separating triangles.
Posted by Kevin Ryde at Dec 18, 2015 6:12 AM.
Line graph of the complete4. Van Rooij and Wilf show this is one of only three line graphs which contain two even triangles with an edge in common. (The other such line graphs are the wheel5 by deleting any vertex of the present graph, and the diamond/kite graph by deleting any 2 adjacent vertices of the present graph leaving two triangles and nothing else.)
A.C.M. van Rooij and H.S. Wilf, "The Interchange Graph of a Finite Graph", Acta Mathematica Academiae Scientiarum Hungaricae, volume 16, 1965, pages 263269.
https://www.math.upenn.edu/~wilf/website/Interchange%20graph.pdf
Posted by Kevin Ryde at Apr 15, 2018 3:55 AM.
This graph has 15 minimal dominating sets which is the most of any n=6 vertices. It is the only n=6 with this many.
Fomin et al report Kratsch noted that taking disjoint copies of this graph gives graphs of (15^(1/6))^n = 1.5704^n many minimal dominating sets. (15^(1/6) is OEIS A011350.)
Fedor V. Fomin, Fabrizio Grandoni, Artem V. Pyatkin, Alexey A. Stepanov, "Combinatorial Bounds via Measure and Conquer: Bounding Minimal Dominating Sets and Applications", ACM Transactions on Algorithms, volume 5, number 1, article 9, November 2008.
http://www.ii.uib.no/~fomin/articles/2008/2008g.pdf
Posted by House of Graphs at Jan 29, 2019 9:24 AM.
A connected integral graph. A graph is called integral if all of its eigenvalues of its adjacency matrix are integral. See "Krzysztof T. ZwierzyĆski, Generating Integral Graphs Using PRACE Research Infrastructure" for more information.
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