On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix

For $e$ a positive integer, we find restrictions modulo $2^e$ on the coefficients of the characteristic polynomial $\chi_S(x)$ of a Seidel matrix $S$. We show that, for a Seidel matrix of order $n$ even (resp. odd), there are at most $2^{\binom{e-2}{2}}$ (resp. $2^{\binom{e-2}{2}+1}$) possibilities for the congruence class of $\chi_S(x)$ modulo $2^e\mathbb Z[x]$. As an application of these results, we obtain an improvement to the upper bound for the number of equiangular lines in $\mathbb R^{17}$, that is, we reduce the known upper bound from $50$ to $49$.Comment: 21 pages, fixed typo in Lemma 2.

Equiangular lines in Euclidean spaces

We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d=14, and d=16. Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.Comment: 24 pages, to appear in JCTA. Corrected an entry in Table

Euler Graphs, Triangle-Free Graphs and Bipartite Graphs in Switching Classes

In the context of graph transformations we look at the operation of switching, which can be viewed as an elegant method for realizing global transformations of graphs through local transformations of the vertices. A switching class is then a set of graphs obtainable from a given start graph by applying the switching operation