Graph details

Graph # 32256

Adjacency matrix

[Too large to display]

Adjacency list

[Too large to display]

HoG graph id

32256

Graph name

Bulgarian Solitaire N=8

Graph submitted by

Kevin Ryde

Invariant values

The definitions of the invariants can be found here.
Invariant Value Invariant Value
Acyclic No Index 2.399
Algebraic Connectivity 0 Laplacian Largest Eigenvalue 5.137
Average Degree 1.909 Longest Induced Cycle 4
Bipartite Yes Longest Induced Path 9
Chromatic Index 3 Matching Number 10
Chromatic Number 2 Maximum Degree 3
Circumference 4 Minimum Degree 1
Claw-Free No Minimum Dominating Set 9
Clique Number 2 Number of Components 2
Connected No Number of Edges 21
Density 0.091 Number of Triangles 0
Diameter infinity Number of Vertices 22
Edge Connectivity 0 Planar Yes
Eulerian No Radius undefined
Genus undefined Regular No
Girth 4 Second Largest Eigenvalue 2.034
Hamiltonian No Smallest Eigenvalue -2.399
Independence Number 12 Vertex Connectivity 0

A table row rendered like this indicates that the graph is marked as being interesting for that invariant.

Comments

Posted by Kevin Ryde at Nov 29, 2018 7:52 AM.
Each vertex is a partition of N=8, ie. a set of integers which sum to 8. There are 22 such. Each edge is from a partition to its successor under the Bulgarian solitaire rule: subtract 1 from each element, sum those 1s as a new term, discard any zeros.

N=8 is the first where the resulting graph is not connected. It has 2 components (OEIS A037306). One ends in a 4-cycle. The other ends in a 2-cycle. That 2-cycle is a single edge in the simple graph here.

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