Graph details

Adjacency matrix
0000100

0000001

0000001

0000010

1000010

0001101

0110010


Adjacency list
1:

5

2:

7

3:

7

4:

6

5:

1
6

6:

4
5
7

7:

2
3
6


HoG graph id
714
Graph name
n/a
Graph submitted by
GraPHedron
Invariant values
The definitions of the invariants can be found
here.
Invariant 
Value 
Invariant 
Value 
Acyclic

Yes

Index

2.053

Algebraic Connectivity

0.322

Laplacian Largest Eigenvalue

4.629

Average Degree

1.714

Longest Induced Cycle

undefined

Bipartite

Yes

Longest Induced Path

4

Chromatic Index

3

Matching Number

3

Chromatic Number

2

Maximum Degree

3

Circumference

undefined

Minimum Degree

1

ClawFree

No

Minimum Dominating Set

3

Clique Number

2

Number of Components

1

Connected

Yes

Number of Edges

6

Density

0.286

Number of Triangles

0

Diameter

4

Number of Vertices

7

Edge Connectivity

1

Planar

Yes

Eulerian

No

Radius

2

Genus

0

Regular

No

Girth

undefined

Second Largest Eigenvalue

1.209

Hamiltonian

No

Smallest Eigenvalue

2.053

Independence Number

4

Vertex Connectivity

1

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 Kevin Ryde at Oct 7, 2018 3:47 AM.
Holton shows that a completely semistable tree is stable, and gives the present tree as an example where the converse does not hold. It is stable, but not completely semistable due to not semistable at the leaf with degree2 neighbour (deleting that introduces new automorphisms). In fact only 2 vertices are semistable, being the degree2 and the degree1 which is distance 2 away from there.
D. A. Holton, "Completely SemiStable Trees", Bulletin of the Australian Mathematical Society, volume 9, issue 3, 1973, pages 355362.
https://doi.org/10.1017/S0004972700043367
Posted by Kevin Ryde at Jan 13, 2019 7:56 AM.
This tree has its independence polynomial and perfect domination polynomial both consisting entirely of real roots. It is one of 2 such trees of n=7 vertices. Their perfect dompolys are the same but indpolys different.
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