Graph details

Graph # 32264

Adjacency matrix

000000001
000000001
000000100
000000010
000000010
000000100
001001001
000110001
110000110

Adjacency list

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

HoG graph id

32264

Graph name

Tree Automorphisms 3 Generators N=9

Graph submitted by

Kevin Ryde

Invariant values

The definitions of the invariants can be found here.
Invariant Value Invariant Value
Acyclic Yes Index 2.288
Algebraic Connectivity 0.268 Laplacian Largest Eigenvalue 5.449
Average Degree 1.778 Longest Induced Cycle undefined
Bipartite Yes Longest Induced Path 4
Chromatic Index 4 Matching Number 3
Chromatic Number 2 Maximum Degree 4
Circumference undefined Minimum Degree 1
Claw-Free No Minimum Dominating Set 3
Clique Number 2 Number of Components 1
Connected Yes Number of Edges 8
Density 0.222 Number of Triangles 0
Diameter 4 Number of Vertices 9
Edge Connectivity 1 Planar Yes
Eulerian No Radius 2
Genus 0 Regular No
Girth undefined Second Largest Eigenvalue 1.414
Hamiltonian No Smallest Eigenvalue -2.288
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 Dec 15, 2018 6:28 AM.
This tree requires at least 3 generators for its automorphism group. Its n=9 vertices is the fewest where a tree requires 3 generators, and this tree is the only such n=9. The automorphism group is (S2 wr S2) x S2, of order 16. S2wrS2 is the dihedral group D4 order 8.

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