# CSASC 2013

9-13 June 2013
Koper, Slovenia
UTC timezone
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# FROM MATRIX TO GRAPH THEORY

Presented by Marko OREL
Type: Oral presentation
Track: Symmetries in Graphs, Maps and Other Discrete Structures

## Content

In matrix theory a preserver problem' demands a characterization of all maps $\Phi$ on a certain set $\mathcal{M}$ of matrices, which leave some function, subset, or relation invariant. In the case a symmetric binary relation $R$ is preserved, such maps $\Phi$ are precisely the endomorphisms of the graph $\Gamma$ with the vertex set $V(\Gamma)=\mathcal{M}$ and the edge set $E(\Gamma)=\{\{m_1,m_2\}\ :\ m_1 R m_2\}$. Hence a preserver problem' demands a characterization of all endomorphisms of the graph $\Gamma$. Often there are some additional assumptions on maps $\Phi$, and we are interested, for example, in characterization of all automorphisms, bijective endomorphisms, etc. In graph theory, a finite graph is a \emph{core}, if all its endomorphisms are automorphisms. In this talk some cores will be presented that arise from matrix structures. These graphs are often highly `symmetrical' and generalize some of the most famous graphs. Moreover, the endomorphisms of some graphs are related to maps that preserve the speed of light on a 4-dimensional Minkowski space-time.

## Place

Location: Koper, Slovenia

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