These notes are not necessarily an accurate representation of what happened in class. The notes written before class say what I think I should say. I sometimes edit the notes after class to make them way what I wish I had said. There may be small mistakes, so I recommend that you check any mathematically precise statement before using it in your own work. These notes were last revised on Septem...

Consider the random matrix obtained from the adjacency matrix of a random d-regular graph by multiplying every entry by a random sign. The largest eigenvalue converges, after proper scaling, to the Tracy–Widom distribution.

We provide a useful method for calculating the state vector of a state equation efficiently in a max-plus algebraic system. For a discrete event system whose precedence relationships are represented by a directed acyclic graph, computing the transition matrix, which includes the Kleene star operation of a weighted adjacency matrix, is occasionally the bottleneck. On the other hand, the common o...

in this paper‎, ‎we define two $n$-square upper hessenberg matrices one of which corresponds to the adjacency matrix of a directed pseudo graph‎. ‎we investigate relations between permanents and determinants of these upper hessenberg matrices‎, ‎and sum formulas of the well-known pell and jacobsthal sequences‎. ‎finally‎, ‎we present two maple 13 procedures in order to calculate permanents of t...

A graph G is said to be nonsingular (resp., singular) if its adjacency matrix A(G) is nonsingular (resp., singular). The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of ±1s. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contract...

In this paper, all connected bipartite graphs are characterized whose third largest Laplacian eigenvalue is less than three. Moreover, the result is used to characterize all connected bipartite graphs with exactly two Laplacian eigenvalues not less than three, and all connected line graphs of bipartite graphs with the third eigenvalue of their adjacency matrices less than one. c © 2003 Elsevier...