# Graph details

 011110 101101 110011 110011 101101 011110

 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

226

Octahedral Graph

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
Claw-Free 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
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.

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), 1875-1891" 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 complete-4. 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 wheel-5 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 263-269.
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. ZwierzyĆski, Generating Integral Graphs Using PRACE Research Infrastructure" for more information.