Graph details

Graph # 32254

Adjacency matrix

00000000001
00000000010
00000000100
00000001000
00000110000
00001000010
00001000100
00010000001
00100010000
01000100001
10000001010

Adjacency list

1: 11
2: 10
3: 9
4: 8
5: 6 7
6: 5 10
7: 5 9
8: 4 11
9: 3 7
10: 2 6 11
11: 1 8 10

HoG graph id

32254

Graph name

Bulgarian Solitaire Tree N=6

Graph submitted by

Kevin Ryde

Invariant values

The definitions of the invariants can be found here.
Invariant Value Invariant Value
Acyclic Yes Index 2.122
Algebraic Connectivity 0.108 Laplacian Largest Eigenvalue 4.698
Average Degree 1.818 Longest Induced Cycle undefined
Bipartite Yes Longest Induced Path 8
Chromatic Index 3 Matching Number 5
Chromatic Number 2 Maximum Degree 3
Circumference undefined Minimum Degree 1
Claw-Free No Minimum Dominating Set 5
Clique Number 2 Number of Components 1
Connected Yes Number of Edges 10
Density 0.182 Number of Triangles 0
Diameter 8 Number of Vertices 11
Edge Connectivity 1 Planar Yes
Eulerian No Radius 4
Genus 0 Regular No
Girth undefined Second Largest Eigenvalue 1.71
Hamiltonian No Smallest Eigenvalue -2.122
Independence Number 6 Vertex Connectivity 1

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 7:43 AM.
Each vertex is a partition of N=6, ie. a set of integers which sum to 6. There are 11 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=6 is a triangular number and the steps form a tree ending at partition 1+2+3 which is unchanged by the solitaire rule and can be reckoned the tree root. This is the leaf nearest the centre.

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