Complex simple Lie algebras are characterised by their Dynkin diagrams and same is true for affine Kac-Moody algebras also. Right angled artin groups, Coxeter groups and many other algebraic structures carries an associated graph and these graphs are concise way to present these groups. My question is what are the other mathematical structures in which there is always an associated graph and the graph reflects many properties of the structure. If anybody can quote explicit results regarding this that will be vey helpful. Thanks a lot for your time. I hope I explained my question better.
Let me try to challenge P. Heinig's quote that there is not a single notable example of a graph-theoretic property precisely corresponding to a notable ring-theoretic property.
Singularities of real analytic curves have associated chord diagrams. Singular points define chord diagrams as follows: desingularization gives a necklace of Moebius bands and annuli. The chord diagram is read on the boundary of the necklace cyclically.
Not all chord diagrams come from analytic curves. Only recently was it understood which chord diagrams arise from singularities of real analytic curves. This happens iff the diagram is collapsable.
Interestingly, there is a pure graph-theoretic characterization of these diagrams. A chord diagram is collapsable if it does not contain subgraphs called houses, gems, dominos or cycles of length greater than 4 (Bendelt-Mulder 1986).
Markov chains in probability theory have graphs associated to them. The graph is finite if the state space is finite. Some properties of the dynamics of the shift on the associated sequence space can be read on the transition matrix or on the graph. For example, the graph is strongly connected iff the matrix is irreducible iff the topological Markov chain is transitive. The mixing property of the shift is equivalent to the connectedness and aperiodicity of the graph.