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.

Another construction of edge-regular graphs with regular cliques

We exhibit a new construction of edge-regular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edge-regular graph with parameters $(24,8,2)$. We also show that edge-regular graphs with $1$-regular cliques that are not strongly regular must have at least $24$ vertices.Comment: 7 page

Real equiangular lines in dimension 18 and the Jacobi identity for complementary subgraphs

We show that the maximum cardinality of an equiangular line system in $\mathbb R^{18}$ is at most $59$. Our proof includes a novel application of the Jacobi identity for complementary subgraphs. In particular, we show that there does not exist a graph whose adjacency matrix has characteristic polynomial $(x-22)(x-2)^{42} (x+6)^{15} (x+8)^2$.Comment: 26 pages. Updated to match the published, journal versio

Equiangular lines in low dimensional Euclidean spaces

We show that the maximum cardinality of an equiangular line system in 14 and 16 dimensions is 28 and 40, respectively, thereby solving a longstanding open problem. We also improve the upper bounds on the cardinality of equiangular line systems in 19 and 20 dimensions to 74 and 94, respectively.Comment: 23 page

Frames over finite fields: Equiangular lines in orthogonal geometry

We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v \leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension~5

Frames over finite fields: Equiangular lines in orthogonal geometry

We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v \leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension~5

The JCMT Gould Belt Survey: Evidence for radiative heating in Serpens MWC 297 and its influence on local star formation

We present SCUBA-2 450micron and 850micron observations of the Serpens MWC 297 region, part of the JCMT Gould Belt Survey of nearby star-forming regions. Simulations suggest that radiative feedback influences the star-formation process and we investigate observational evidence for this by constructing temperature maps. Maps are derived from the ratio of SCUBA-2 fluxes and a two component model of the JCMT beam for a fixed dust opacity spectral index of beta = 1.8. Within 40 of the B1.5Ve Herbig star MWC 297, the submillimetre fluxes are contaminated by free-free emission with a spectral index of 1.03+-0.02, consistent with an ultra-compact HII region and polar winds/jets. Contamination accounts for 73+-5 per cent and 82+-4 per cent of peak flux at 450micron and 850micron respectively. The residual thermal disk of the star is almost undetectable at these wavelengths. Young Stellar Objects are confirmed where SCUBA-2 850micron clumps identified by the fellwalker algorithm coincide with Spitzer Gould Belt Survey detections. We identify 23 objects and use Tbol to classify nine YSOs with masses 0.09 to 5.1 Msun. We find two Class 0, one Class 0/I, three Class I and three Class II sources. The mean temperature is 15+-2K for the nine YSOs and 32+-4K for the 14 starless clumps. We observe a starless clump with an abnormally high mean temperature of 46+-2K and conclude that it is radiatively heated by the star MWC 297. Jeans stability provides evidence that radiative heating by the star MWC 297 may be suppressing clump collapse.Comment: 24 pages, 13 figures, 7 table