University of Minnesota Combinatorics Seminar: 2020-2021

Date: 02/05/2021

Speaker: Rob Davis

Analyzing Power Grid Stability via the Dragon Marriage Theorem
One real-life situation application of graph theory is the study of electrical grids: they have to be constructed carefully since unstable grids can lead to brownouts, blackouts, damaged equipment, or other possible problems. If we know the connections in the grid that we want, how can the voltages at each node be coordinated in a way that makes sure the network stays stable? This is a difficult question, but even knowing the number of ways to keep a network stable can help. In this talk, we will see how to count the number of "stable solutions" using discrete geometric and algebraic methods. These methods will help us obtain recurrences for networks satisfying mild conditions. Consequently, we obtain explicit, non-recursive formulas for the number of stable solutions for a large class of outerplanar graphs, and conjecture that the formula holds for all outerplanar graphs. Key to these results is the Dragon Marriage Theorem: a generalization of Hall's Matching Theorem with far-reaching implications.

Analyzing Power Grid Stability via the Dragon Marriage Theorem

One real-life situation application of graph theory is the study of electrical grids: they have to be constructed carefully since unstable grids can lead to brownouts, blackouts, damaged equipment, or other possible problems. If we know the connections in the grid that we want, how can the voltages at each node be coordinated in a way that makes sure the network stays stable? This is a difficult question, but even knowing the number of ways to keep a network stable can help. In this talk, we will see how to count the number of "stable solutions" using discrete geometric and algebraic methods. These methods will help us obtain recurrences for networks satisfying mild conditions. Consequently, we obtain explicit, non-recursive formulas for the number of stable solutions for a large class of outerplanar graphs, and conjecture that the formula holds for all outerplanar graphs. Key to these results is the Dragon Marriage Theorem: a generalization of Hall's Matching Theorem with far-reaching implications.