Mike Krebs

On the spectra of Johnson graphs

The spectrum of a Johnson graph is known to be given by the Eberlein polynomial. In this paper, a straightforward representation-theoretic derivation of this fact is presented. Also discussed are some consequences of this formula, such as the fact that infinitely many of them are Ramanujan.

Zero Sums on Unit Square Vertex Sets and Plane Colorings

Abstract We prove that if a real-valued function of the plane sums to zero on the four vertices of every unit square, then it must be the zero function. This fact implies a lower bound in a “coloring of the plane” problem similar to the famous Hadwiger–Nelson problem, which asks for the smallest number of colors needed to assign every point in the plane a color so that no two points of unit distance apart have the same color.

The Combinatorial Trace Method in Action.

Summary On any finite graph, the number of closed walks of length k is equal to the sum of the kth powers of the eigenvalues of any adjacency matrix. This simple observation is the basis for the combinatorial trace method, wherein we attempt to count (or bound) the number of closed walks of a given length so as to obtain information about the graphs eigenvalues, and vice versa. We give a brief overview and present some simple but interesting examples. The method is also the source of interesting, accessible undergraduate projects.

Can a Subset's Topology Detect Continuous Extensions?

Let Q denote the set of rational numbers and R the set of reals. Consider the function k : Q → Q defined by k(x) = 0 if x < √2 and k(x) = 1 if x > √2. A standard exercise in an introductory point-set topology class is to show that k is continuous but does not extend continuously to a function from R to R. Here, we topologize Q in the standard way, i.e., as a subspace of R where R has the Euclidean topology. One way to think of this is that the standard topology on Q does not “detect” continuous extensions to R. Can any topology on Q do so? More precisely: