On a Class of Algebraically Defined Graphs

Let denote be the n-dimensional vector spaces over a field F. For and each , let be a function of 2i variables. We consider a bipartite graph whose vertex partitions P and L are copies of with and being joined by an edge if and only if the following n-1 equalities are satisfied:

I this talk we survey some known and some new results on such graphs based on the work of V.A. Ustimenko, A. J. Woldar and the speaker.

Keywords: forbidden cycles, girth, graph homomorphism, eigenvalue, edge decomposition.