# Core (graph theory)

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In the mathematical field of graph theory, a core is a notion that describes behavior of a graph with respect to graph homomorphisms.

## Definition

Graph ${\displaystyle G}$ is a core if every homomorphism ${\displaystyle f:G\to G}$ is an isomorphism, that is it is a bijection of vertices of ${\displaystyle G}$.

A core of a graph ${\displaystyle H}$ is a graph ${\displaystyle G}$ such that

1. There exists a homomorphism from ${\displaystyle H}$ to ${\displaystyle G}$,
2. there exists a homomorphism from ${\displaystyle G}$ to ${\displaystyle H}$, and
3. ${\displaystyle G}$ is minimal with this property.

Two graphs are said to be homomorphism equivalent or hom-equivalent if they have isomorphic cores.

## Examples

• Any complete graph is a core.
• A cycle of odd length is its own core.
• Every two cycles of even length, and more generally every two bipartite graphs are hom-equivalent. The core of each of these graphs is the two-vertex complete graph K2.

## Properties

Every graph has a core, which is determined uniquely, up to isomorphism. The core of a graph G is always an induced subgraph of G. If ${\displaystyle G\to H}$ and ${\displaystyle H\to G}$ then the graphs ${\displaystyle G}$ and ${\displaystyle H}$ are necessarily hom-equivalent.

## References

• Godsil, Chris, and Royle, Gordon. Algebraic Graph Theory. Graduate Texts in Mathematics, Vol. 207. Springer-Verlag, New York, 2001. Chapter 6 section 2.
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