Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1604.03423

Graph Matrices: Norm Bounds and Applications

Kwangjun Ahn·Dhruv Medarametla·Aaron Potechin
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we derive nearly tight probabilistic norm bounds for a class of random matrices we call graph matrices. While the classical case of symmetric matrices with independent random entries (Wigner's matrices) is a special case, in general, the entries of our matrices will be dependent in a way that can be specified in terms of a fixed-size graph we refer to as the shape. For Wigner's matrices, this shape is $K_2$, the clique on 2 vertices. To prove our norm bounds, we use the trace power method. In a recent series of papers by Potechin and coauthors, graph matrices played a crucial role in proving average-case lower bounds for the Sum-of-Squares (SoS) hierarchy of proof systems, one of the most powerful, but difficult to analyze, techniques in combinatorial optimization. In particular, graph matrices played a crucial role in proving that low-degree SoS cannot refute the existence of a large clique in a random graph and proving that low-degree SoS cannot prove a tight lower bound on the ground state energy of the Sherrington-Kirkpatrick Hamiltonian. In this paper, we give several additional applications of graph matrices. We show that for several technical lemmas in the literature, while the original analyses were quite involved, we can give direct proofs using graph matrices and our norm bounds.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kwangjun Ahn, Dhruv Medarametla, Aaron Potechin. 2021-04-29. Graph Matrices: Norm Bounds and Applications. https://arxiv.org/abs/1604.03423

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Orthogonal Pairs in Maps from the Sphere to the Circle

We prove that, for any $f:S^2\to S^1$ and any $\varepsilon>0$, there exist orthogonal vectors $x,y\in S^2$ such that the length of the shortest arc between $f(x)$ and $f(y)$ is at most $π/2 +\varepsilon$. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.

math.CO↗
arXiv2026

Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles

For a dissection $D$ of the unit square into $n$ nondegenerate triangles, let $R(D)=\max_i a_i-\min_i a_i, Δ(n)=\inf_D R(D).$ We prove that this infimum is attained for every $n\ge2$, and determine the exact minima for $n=5$ and $n=7$, allowing T-junctions. For five triangles, $Δ(5)=\frac{5\sqrt5-11}{8};$ equality holds precisely when three areas equal $(3-\sqrt5)/4$ and two equal $(3\sqrt5-5)/8$. For seven triangles, $Δ(7)=r_7$, where $r_7$ is the unique root in $(0,1/4900)$ of $864r^4+2160r^3-6060r^2+4972r-1.$ Every minimizer has four areas $(1+3r_7)/7$ and three areas $(1-4r_7)/7$, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound $Δ(9)\le 0.0001273496861283553341\ldots,$ which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than $1/3500$; hence a global minimizer must lie outside that family. The exact value of $Δ(9)$ remains open.

math.CO↗
arXiv2026

Chromatic symmetric functions for annular webs

We introduce a combinatorial definition of chromatic symmetric functions for annular webs. We prove their symmetry by constructing a web analogue of the Shareshian--Wachs involution and show that they coincide with the symmetric functions associated to annular webs via Turaev's isomorphism. We then derive explicit formulas for their hook Schur coefficients. We also introduce web LLT functions, whose hook Schur coefficients admit positive Laurent-polynomial formulas. These formulas yield a combinatorial expression for the coefficients of the HOMFLY--PT polynomial of an annular web.

math.CO↗