Mathematical Sciences Department
Discrete Math Seminar
Speaker: Ralihe Raul Villagran, WPI
Monday, November 13, 2023
3:00 pm - 3:50 pm
Salisbury Labs 407
Title: Arithmetical structures of graphs
Abstract: Let G be a graph and A(G) its adjacency matrix. Then a pair of positive integer vectors (d,r) is an arithmetical structure of G if A(G)d=d○r, where ○ is the entry-wise product, and r is a primitive integer vector. The set of arithmetical structures of a (directed) graph without loops is finite if and only if the graph is (strongly) connected. Therefore the following natural questions follow: How many arithmetical structures can a graph have? Can we characterize the sets of arithmetical structures? For instance, it is known that the number of arithmetical structures of the path with n+1 vertices is the n-th Catalan number. On the other hand, the arithmetical structures of the complete describes the problem of finding egyptians fractions. We will review these and further results, as well as open problems.