We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott’s code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.

The SIC Question: History and State of Play

Signed graphs are graphs whose edges get a sign +1 or −1 (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs.
Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs.

/pdf/open-problems-in-the-spectral-theory-of-signed-graphs-1fc486xlqt.pdf

Open problems in the spectral theory of signed graphs

Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.

Introduction to Matrix Analysis. By Richard Bellman. 2nd Edition. Pp. xxiii, 403. £7·50. (McGraw-Hill.)

/pdf/the-journey-of-the-union-closed-sets-conjecture-7edcsph88t.pdf

The Journey of the Union-Closed Sets Conjecture

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red–blue colouring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If a graph G is connected, it is well known and easy to show that R(G, H) ≥ (|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ(H) is the size of the smallest colour class in a χ(H)-colouring of H. A graph G is called H-good if R(G, H) = (|G|−1)(χ(H)−1)+σ(H). The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then.In this paper we show that if n≥ Ω(|H| log4 |H|) then every n-vertex bounded degree tree T is H-good. The dependency between n and |H| is tight up to log factors. This substantially improves a result of Erdős, Faudree, Rousseau, and Schelp from 1985, who proved that n-vertex bounded degree trees are H-good when n ≥ Ω(|H|4).

/pdf/ramsey-goodness-of-bounded-degree-trees-12rxiciiew.pdf

Ramsey Goodness of Bounded Degree Trees

The minrank of a graph G on the set of vertices [n] over a field $$\mathbb{F}$$ is the minimum possible rank of a matrix $$M \in \mathbb{F}^{{n}\times{n}}$$ with nonzero diagonal entries such that Mi,j = 0 whenever i and j are distinct nonadjacent vertices of G. This notion, over the real field, arises in the study of the Lovasz theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph G(n, p) over any finite or infinite field, showing that for every field $$\mathbb{F}=\mathbb{F}(n)$$ and every p = p(n) satisfying n-1 ≤ p ≤ 1 - n-0.99, the minrank of G = G(n, p) over $$\mathbb{F}$$ is $$\Theta(\frac{n {\rm{log}}(1/p)}{{\rm{log}} n})$$ with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev and Weinstein, who proved the above result for finite fields of size at most nO(1), with tools from linear algebra, including an estimate of Ronyai, Babai and Ganapathy for the number of zero-patterns of a sequence of polynomials.

The minrank of random graphs over arbitrary fields

Let F be a finite union-closed family of sets whose largest set con- tains n elements In (6), Wojcik defined the density of F to be the ratio of the average set size of F to n and conjectured that the mini- mum density over all union-closed families whose largest set contains n elements is (1+o(1))log 2 n/(2n) as n → ∞ We use a result of Reimer (3) to show that the density of F is always at least log 2 n/(2n), verify- ing Wojcik's conjecture As a corollary we show that for n ≥ 16, some element must appear in at least p (log 2 n)/n(|F|/2) sets of F

Minimum density of union-closed families

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle $\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in $\mathbb{R}^n$ with common angle $\theta$. Moreover, this is achievable only for $\theta = \arccos(1/3)$. We also show that for any set of $k$ fixed angles, one can find at most $O(n^k)$ lines in $\mathbb{R}^n$ having these angles. This bound, conjectured by Bukh, substantially improves the estimate of Delsarte, Goethals and Seidel from 1975. Various extensions of these results to the more general setting of spherical codes will be discussed as well.

Equiangular Lines and Spherical Codes in Euclidean Space

Let $$x_1, \ldots , x_n \in \mathbb {R}^d$$
 be unit vectors such that among any three there is an orthogonal pair. How large can n be as a function of d, and how large can the length of $$x_1 + \cdots + x_n$$
 be? The answers to these two celebrated questions, asked by Erdős and Lovasz, are closely related to orthonormal representations of triangle-free graphs, in particular to their Lovasz $$\vartheta $$
 -function and minimum semidefinite rank. In this paper, we study these parameters for general H-free graphs. In particular, we show that for certain bipartite graphs H, there is a connection between the Turan number of H and the maximum of $$\vartheta (\overline{G})$$
 over all H-free graphs G.

Orthonormal Representations of H-Free Graphs

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