Related papers: The asymptotic spectrum of graphs and the Shannon …
We study quantum versions of the Shannon capacity of graphs and non-commutative graphs. We introduce the asymptotic spectrum of graphs with respect to quantum and entanglement-assisted homomorphisms, and we introduce the asymptotic spectrum…
The asymptotic spectrum of graphs, introduced by Zuiddam (arXiv:1807.00169, 2018), is the space of graph parameters that are additive under disjoint union, multiplicative under the strong product, normalized and monotone under homomorphisms…
Determining the Shannon capacity of graphs is a long-standing open problem in information theory, graph theory and combinatorial optimization. Over decades, a wide range of upper and lower bound methods have been developed to analyze this…
The Shannon capacity of graphs, introduced by Shannon in 1956 to model zero-error communication, asks for determining the rate of growth of independent sets in strong powers of graphs. Much is still unknown about this parameter, for…
We continue the study of the quantum channel version of Shannon's zero-error capacity problem. We generalize the celebrated Haemers bound to noncommutative graphs (obtained from quantum channels). We prove basic properties of this bound,…
For a graph $G$, its $k$-th graph power $G^k$ is constructed by placing an edge between two vertices if they are within distance $k$. We consider the problem of deriving upper bounds on the Shannon capacity of graph powers by using spectral…
This paper delves into three research directions, leveraging the Lov\'{a}sz $\vartheta$-function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses…
We investigate the effect of the well-known Mycielski construction on the Shannon capacity of graphs and on one of its most prominent upper bounds, the (complementary) Lov\'asz theta number. We prove that if the Shannon capacity of a graph,…
In this note, we present a fractional version of Haemers' bound on the Shannon capacity of a graph, which is originally due to Blasiak. This bound is a common strengthening of both Haemers' bound and the fractional chromatic number of a…
In this paper, we give spectral upper bounds for the independence number of even uniform hypergraphs and graphs, extend the Hoffman bound to even uniform hypergraphs, and give a simple spectral condition for determining the independence…
The zero-error capacity of a classical channel is a parameter of its confusability graph, and is equal to the minimum of the values of graph parameters that are additive under the disjoint union, multiplicative under the strong product,…
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…
Given a semiring with a preorder subject to certain conditions, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is a compact Hausdorff space together with a map from the semiring to the ring of continuous…
We prove an upper bound on the Shannon capacity of a graph via a linear programming variation. We show that our bound can outperform both the Lov\'asz theta number and the Haemers minimum rank bound. As a by-product, we also obtain a new…
Motivated by the problem of zero-error broadcasting, we introduce a new notion of graph capacity, termed $\rho$-capacity, that generalizes the Shannon capacity of a graph. We derive upper and lower bounds on the $\rho$-capacity of arbitrary…
We introduce a generalization of the celebrated Lov\'asz theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial…
Non-commutative graph theory is an operator space generalization of graph theory. Well known graph parameters such as the independence number and Lov\'asz theta function were first generalized to this setting by Duan, Severini, and Winter.…
We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in [Kennedy et al, Calc. Var. 60 (2021), 61]. We provide sharp lower and upper estimates for minimal partition energies in…
We develop a group-theoretic approach to the Shannon capacity problem. Using this approach we extend and recover, in a structured and unified manner, various families of previously known lower bounds on the Shannon capacity. Bohman (2003)…
In a recent work on the bipartite Erd\H{o}s-R\'{e}nyi graph, Do et al. (2023) established upper bounds on the number of connected labeled bipartite graphs with a fixed surplus. We use some recent encodings of bipartite random graphs in…