Title: Higher dimensional graph theory.

Abstract: Many concepts of the graphs theory have topological nature. As such they admit higher 
dimensional generalizations to simplicial and cell complexes. In particular, a concept of 
spanning subgraph and classical Cayley’s formula for the number of spanning subgraphs of a 
complete graph was generalized to higher dimension by G.Kalai in 1983. Then in 2009 
A.Duval, C.Klivans, and J.Martin generalize the matrix-tree theorem to cellular complexes. 
Another such concept is the notion of polynomial invariant of a graph such as chromatic, flow, 
and the Tutte polynomials. The higher dimensional generalization of the flow polynomial was 
introduced by R.Bott in 1952 without any relation with the graph theory and then they were 
studied by Z.Wang in 1994. More recently, in 2010, V.Krushkal and D.Renardy suggested a 
generalization of the Tutte polynomial to cell complexes. I will speak about these 
generalizations and their mutual relationships.