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# Diamond graph

Diamond graph
Vertices4
Edges5
Radius1
Diameter2
Girth3
Automorphisms4 (Klein four-group: ${\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /2\mathbb {Z} )}$
Chromatic number3
Chromatic index3
PropertiesHamiltonian
Planar
Unit distance
Table of graphs and parameters

In the mathematical field of graph theory, the diamond graph is a planar, undirected graph with 4 vertices and 5 edges.[1][2] It consists of a complete graph ${\displaystyle K_{4}}$ minus one edge.

The diamond graph has radius 1, diameter 2, girth 3, chromatic number 3 and chromatic index 3. It is also a 2-vertex-connected and a 2-edge-connected, graceful,[3] Hamiltonian graph.

## Diamond-free graphs and forbidden minor

A graph is diamond-free if it has no diamond as an induced subgraph. The triangle-free graphs are diamond-free graphs, since every diamond contains a triangle. The diamond-free graphs are locally clustered: that is, they are the graphs in which every neighborhood is a cluster graph. Alternatively, a graph is diamond-free if and only if every pair of maximal cliques in the graph shares at most one vertex.

The family of graphs in which each connected component is a cactus graph is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor. This minor is the diamond graph.[4]

If both the butterfly graph and the diamond graph are forbidden minors, the family of graphs obtained is the family of pseudoforests.

## Algebraic properties

The full automorphism group of the diamond graph is a group of order 4 isomorphic to the Klein four-group, the direct product of the cyclic group ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ with itself.

The characteristic polynomial of the diamond graph is ${\displaystyle x(x+1)(x^{2}-x-4)}$. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.

## References

1. ^ Weisstein, Eric W. "Diamond Graph". MathWorld.
2. ^ ISGCI: Information System on Graph Classes and their Inclusions "List of Small Graphs".
3. ^ Sin-Min Lee, Y.C. Pan and Ming-Chen Tsai. "On Vertex-graceful (p,p+l)-Graphs". [1] Archived 2008-08-07 at the Wayback Machine
4. ^ El-Mallah, Ehab; Colbourn, Charles J. (1988), "The complexity of some edge deletion problems", IEEE Transactions on Circuits and Systems, 35 (3): 354–362, doi:10.1109/31.1748.