Using group theory Algebraic graph theory

a cayley graph alternating group a4, forming truncated tetrahedron in 3 dimensions. cayley graphs vertex-transitive, vertex-transitive graphs (like petersen graph) not cayley graphs.



a proper vertex coloring of petersen graph 3 colors, minimum number possible. according chromatic polynomial, there 120 such colorings 3 colors.

this second branch of algebraic graph theory related first, since symmetry properties of graph reflected in spectrum. in particular, spectrum of highly symmetrical graph, such petersen graph, has few distinct values (the petersen graph has 3, minimum possible, given diameter). cayley graphs, spectrum can related directly structure of group, in particular irreducible characters.