The asymptotic spectrum of graphs and the Shannon capacity

We introduce the asymptotic spectrum of graphs and apply the theory of asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new dual characterisation of the Shannon capacity of graphs. Elements in the asymptotic spectrum of graphs include the Lov\'asz theta number, the fractional clique cover number, the complement of the fractional orthogonal rank and the fractional Haemers bounds

Relative Fractional Independence Number and Its Applications

We define the relative fractional independence number of two graphs, $G$ and $H$, as $$\alpha^*(G|H)=\max_{W}\frac{\alpha(G\boxtimes W)}{\alpha(H\boxtimes W)},$$ where the maximum is taken over all graphs $W$, $G\boxtimes W$ is the strong product of $G$ and $W$, and $\alpha$ denotes the independence number. We give a non-trivial linear program to compute $\alpha^*(G|H)$ and discuss some of its properties. We show that $$\alpha^*(G|H)\geq \frac{X(G)}{X(H)},$$ where $X(G)$ can be the independence number, the zero-error Shannon capacity, the fractional independence number, the Lov'{a}sz number, or the Schrijver's or Szegedy's variants of the Lov'{a}sz number of a graph $G$. This inequality is the first explicit non-trivial upper bound on the ratio of the invariants of two arbitrary graphs, as mentioned earlier, which can also be used to obtain upper or lower bounds for these invariants. As explicit applications, we present new upper bounds for the ratio of the zero-error Shannon capacity of two Cayley graphs and compute new lower bounds on the Shannon capacity of certain Johnson graphs (yielding the exact value of their Haemers number). Moreover, we show that the relative fractional independence number can be used to present a stronger version of the well-known No-Homomorphism Lemma. The No-Homomorphism Lemma is widely used to show the non-existence of a homomorphism between two graphs and is also used to give an upper bound on the independence number of a graph. Our extension of the No-Homomorphism Lemma is computationally more accessible than its original version

On a tracial version of Haemers bound

We extend upper bounds on the quantum independence number and the quantum Shannon capacity of graphs to their counterparts in the commuting operator model. We introduce a von Neumann algebraic generalization of the fractional Haemers bound (over $\mathbb{C}$) and prove that the generalization upper bounds the commuting quantum independence number. We call our bound the tracial Haemers bound, and we prove that it is multiplicative with respect to the strong product. In particular, this makes it an upper bound on the Shannon capacity. The tracial Haemers bound is incomparable with the Lov\'asz theta function, another well-known upper bound on the Shannon capacity. We show that separating the tracial and fractional Haemers bounds would refute Connes' embedding conjecture. Along the way, we prove that the tracial rank and tracial Haemers bound are elements of the (commuting quantum) asymptotic spectrum of graphs (Zuiddam, Combinatorica, 2019). We also show that the inertia bound (an upper bound on the quantum independence number) upper bounds the commuting quantum independence number.Comment: 39 pages, comments are welcom

Graph Theory versus Minimum Rank for Index Coding

We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic factor from the chromatic number. This is in striking contrast to minrank since prior work has shown that it can outperform the chromatic number by a polynomial factor in some cases. The conclusion is that all known graph theoretic bounds are not much stronger than the chromatic number.Comment: 8 pages, 2 figures. Submitted to ISIT 201

A unified construction of semiring-homomorphic graph invariants

It has recently been observed by Zuiddam that finite graphs form a preordered commutative semiring under the graph homomorphism preorder together with join and disjunctive product as addition and multiplication, respectively. This led to a new characterization of the Shannon capacity $\Theta$ via Strassen's Positivstellensatz: $\Theta(\bar{G}) = \inf_f f(G)$, where $f : \mathsf{Graph} \to \mathbb{R}_+$ ranges over all monotone semiring homomorphisms. Constructing and classifying graph invariants $\mathsf{Graph} \to \mathbb{R}_+$ which are monotone under graph homomorphisms, additive under join, and multiplicative under disjunctive product is therefore of major interest. We call such invariants semiring-homomorphic. The only known such invariants are all of a fractional nature: the fractional chromatic number, the projective rank, the fractional Haemers bounds, as well as the Lov\'asz number (with the latter two evaluated on the complementary graph). Here, we provide a unified construction of these invariants based on linear-like semiring families of graphs. Along the way, we also investigate the additional algebraic structure on the semiring of graphs corresponding to fractionalization. Linear-like semiring families of graphs are a new concept of combinatorial geometry different from matroids which may be of independent interest.Comment: 25 pages. v3: incorporated referee's suggestion