I will introduce network matrices of (oriented) graphs and binet and tour matrices of bidirected graphs (oriented signed graphs), as well as the operations of 1-, 2-, and 3-summation on matrices. I will discuss the connections between tour matrices and bidirected graphs and present a new result that tour matrices are closed under 1-, 2-, and 3-summation, along with the flavor of the proof (time permitting).
This is work of Appa, Kotnyek, Papalamprou, and Pitsoulis.