Graph details

Graph # 714

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
Claw-Free 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), 1875-1891" for more information.

Posted by Kevin Ryde at Oct 7, 2018 3:47 AM.
Holton shows that a completely semi-stable tree is stable, and gives the present tree as an example where the converse does not hold. It is stable, but not completely semi-stable due to not semi-stable at the leaf with degree-2 neighbour (deleting that introduces new automorphisms). In fact only 2 vertices are semi-stable, being the degree-2 and the degree-1 which is distance 2 away from there.

D. A. Holton, "Completely Semi-Stable Trees", Bulletin of the Australian Mathematical Society, volume 9, issue 3, 1973, pages 355-362.
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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