University of Manitoba

Kemeny's constant, a fundamental parameter in the theory of Markov chains, has recently received significant attention within the graph theory community. Originally defined for a discrete, finite, time-homogeneous, and irreducible Markov chain based...

In this talk, I will discuss certain extremal configurations in Q-polynomial distance-regular graphs (or P- and Q-polynomial association schemes), which we call descendents. The descendents of a hypercube are precisely its faces. I will show how they...

Biological systems often give rise to systems of first order ordinary differential equations (ODEs). Linearization then yields a system x' = Ax, where A is the community matrix. By contrast, mechanical systems often give rise to a second order ODE...

We first review some known results about resistance matrix of an undirected graph. We then define resistance distance in a strongly connected, directed graph. When the graph is balanced, that is, the indegree and the outdegree of any vertex are the...

After a review of n-by-n permutation matrices and n-by-n alternating sign matrices (ASMs), and their relationship, we will discuss the cones that they generate and their Hilbert Bases.

We often encounter matrices whose pattern (zero-nonzero, or sign) is known while the precise value of each entry is not clear. Thus, a natural question is what we can say about the spectral property of matrices of a given pattern. When the matrix is...

The eccentricity matrix ε(G) of a connected graph G is obtained from the distance matrix of G by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of ε(G) are the ε-eigenvalues...

Graphs having three distinct eigenvalues are a fundamental object of study in spectral graph theory. Strongly regular graphs are the most well-studied examples. In 1995, at the 15th British Combinatorial Conference, Willem Haemers asked do there...

Let G be a connected simple graph and L be its Laplacian matrix. Let v be a cut vertex and B be a branch at v. Assume that v_1 is the only vertex in B adjacent to v. Let P be a path that starts at v_1 and stays inside B. It is shown that the...

Suppose Alice can send a quantum message to Bob in a quantum system modeled as a finite graph. But, imagine there is an eavesdropper which can attach an infinite-dimensional quantum system to the finite graph used by Alice and Bob. Can Alice still...