Three short proofs in graph theory

Two Shorter Proofs in Spectral Graph Theory

Let G be a simple graph with V (E) as its vertex (respectively, edge) set. The spectrum of G is the spectrum of its adjacency matrix. For all other definitions (or notation not given here), especially those related to graph spectra, see [1]. The purpose of this note is to provide shorter proofs of two inequalities already known in spectral graph theory. The first bounds vertex eccentricities of...

Three conjectures in extremal spectral graph theory

We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P2 and Pn−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetković and Rowlinson from 1...

New Proofs in Graph Minors

Spectral graph theory: Three common spectra∗

In this first talk we will introduce three of the most commonly used types of matrices in spectral graph theory. They are the adjacency matrix, the combinatorial Laplacian, and the normalized Laplacian. We also will give some simple examples of how the spectrum can be used for each of these types.

Short Discreet Proofs

We show how to produce short proofs of theorems such that a distrusting Verifier can be convinced that the theorem is true yet obtains no information about the proof itself. The proofs are non-interactive provided that the quadratic residuosity bit commitment scheme is available to the Prover and Verifier. For typical applications, the proofs are short enough to fit on a floppy disk.