On McKay's propagation theorem for the Foulkes conjecture

We translate the main theorem in Tom McKay's paper "On plethysm conjectures of Stanley and Foulkes" (J. Alg. 319, 2008, pp. 2050-2071) to the language of weight spaces and projections onto invariant spaces of tensors, which makes its proof short and elegant.Comment: 9 page

Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group

We show the Alon-Tarsi conjecture on Latin squares is equivalent to a very special case of a conjecture made independently by Hadamard and Howe, and to the non-vanishing of some interesting integrals over SU(n). Our investigations were motivated by geometric complexity theory.Comment: 7 page

Singular graphs

Let Γ be a simple graph on a finite vertex set V and let A be its adjacency matrix. Then Γ is said to be singular if and only if 0 is an eigenvalue of A. The nullity (singularity) of Γ, denoted by null(Γ), is the algebraic multiplicity of the eigenvalue 0 in the spectrum of Γ. In 1957, Collatz and Sinogowitz [57] posed the problem of characterizing singular graphs. Singular graphs have important applications in mathematics and science. In chemistry the importance of singular graphs lies in the fact that a singular molecular graph, with vertices formed by atoms, edges corresponding to bonds between the atoms in the molecule, often is associated to compounds that are more reactive or unstable. By this reason, the chemists have a great interest in this problem. The general problem of characterising singular graphs is easy to state but it seems too difficult at this time. In this work, we investigate this problem for graphs in general and graphs with a vertex transitive group G of automorphisms. In some cases we determine the nullity of such graphs. We characterize singular Cayley graphs over cyclic groups. We show that vertex transitive graphs where |V| is prime are non-singular. The relationship between the irreducible representations of G and the eigenspaces of Γ is studied