On flows of graphs

Tutte\u27s 3-flow Conjecture, 4-flow Conjecture, and 5-flow Conjecture are among the most fascinating problems in graph theory. In this dissertation, we mainly focus on the nowhere-zero integer flow of graphs, the circular flow of graphs and the bidirected flow of graphs. We confirm Tutte\u27s 3-flow Conjecture for the family of squares of graphs and the family of triangularly connected graphs. In fact, we obtain much stronger results on this conjecture in terms of group connectivity and get the complete characterization of such graphs in those families which do not admit nowhere-zero 3-flows. For the circular flows of graphs, we establish some sufficient conditions for a graph to have circular flow index less than 4, which generalizes a new known result to a large family of graphs. For the Bidirected Flow Conjecture, we prove it to be true for 6-edge connected graphs

Flows on Bidirected Graphs

The study of nowhere-zero flows began with a key observation of Tutte that in planar graphs, nowhere-zero k-flows are dual to k-colourings (in the form of k-tensions). Tutte conjectured that every graph without a cut-edge has a nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero 6-flow. For a graph embedded in an orientable surface of higher genus, flows are not dual to colourings, but to local-tensions. By Seymour's theorem, every graph on an orientable surface without the obvious obstruction has a nowhere-zero 6-local-tension. Bouchet conjectured that the same should hold true on non-orientable surfaces. Equivalently, Bouchet conjectured that every bidirected graph with a nowhere-zero $\mathbb{Z}$-flow has a nowhere-zero 6-flow. Our main result establishes that every such graph has a nowhere-zero 12-flow.Comment: 24 pages, 2 figure

Positivity for Gaussian graphical models

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of nonzero subdeterminants.Comment: 16 pages, 3 figure