Apr 20th, 1:00 PM - 4:00 PM
Counting Walks From Distinct Vertices on Directed and Undirected Graphs
Dr. Uwe Kaiser
In this talk we will explore the idea of counting subgraphs with special properties of a given graph. The subgraphs we will focus on are trees with specified leaves, and we will look into algorithms well understood in the literature, as well as some we will build, to convert the trees of our interest into algebraic objects like sequences. In the case of only two leaves we study simple paths between two given vertices. We will then be able to distinguish particular patterns in the Adjacency Matrix, and other useful matrices, given from the graphs we study, which will allow us to produce counting procedures, similar to the Matrix Tree Theorem, to count the trees we are interested in. First we will look into undirected graphs and use Prufer’s method as a guide to developing our own method for assigning the trees of our interest to sequences that are special to them. Next, we will use the Matrix Tree Theorem as a guide to develop a counting scheme to determine the number of trees of our interest that exist in a connected and undirected graph. Lastly we will expand our research to directed graphs, counting trees with our desired properties in those graphs. Applications of this research include but are not limited to determining the number of different driving directions a GPS navigator can use to guide drivers to their destinations, and with the use of weighting functions give optimizations for fuel economy, driving time, distance, et cetera. If time allows we will explore the effect of these weighting functions and use Kruskal’s Algorithm as a guide to determining the optimal walk along our trees in the newly weighted graphs.