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 graph. We show that this fractional version outperforms any bound on the Shannon capacity that could be attained through Haemers' bound. We show also that this ...

We tackle the problem of estimating the Shannon capacity of cycles of odd length. We present some strategies which allow us to nd tight bounds on the Shannon capacity of cycles of various odd lengths, and suggest that the diiculty of obtaining a general result may be related to diierent behaviours of the capacity, depending on the \structure" of the odd integer representing the cycle length. We...

Shannon OR-capacity $C_{\rm OR}(G)$ of a graph $G$, that is the traditionally more often used AND-capacity complementary graph, homomorphism monotone parameter therefore OR}(F\times G)\leqslant\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ holds for every pair graphs, where $F\times G$ categorical product graphs $F$ and $G$. Here we initiate study question when could expect equality in this inequality. U...

Despite much impressive work (e.g. [1], [3], [4], [5], [7]) since the introduction of the Shannon capacity in [8], many natural questions regarding this parameter remain widely open (see [2], [6] for surveys). Let G1 + G2 denote the disjoint union of the graphs G1 and G2. It is easy to see that c(G1 + G2) ≥ c(G1) + c(G2). Shannon [8] conjectured that c(G1 + G2) = c(G1) + c(G2), but this was dis...

A biclique in a graph Γ is a complete bipartite subgraph of Γ. We give bounds for the maximum number of edges in a biclique in terms of the eigenvalues of matrix representations of Γ. These bounds show a strong similarity with Lovász’s bound θ(Γ) for the Shannon capacity of Γ. Motivated by this similarity we investigate bicliques and the bounds in certain product graphs.

Motivated by the problem of zero-error broadcasting, we introduce a new notion of graph capacity, termed ρ-capacity, that generalizes the Shannon capacity of a graph. We derive upper and lower bounds on the ρ-capacity of arbitrary graphs, and provide a Lovász-type upper bound for regular graphs. We study the behavior of the ρ-capacity under two graph operations: the strong product and the disjo...

In this paper we consider a wiretap channel with a secret key buffer. We use the coding scheme of [1] to enhance the secrecy rate to the capacity of the main channel, while storing each securely transmitted message in the secret key buffer. We use the oldest secret bits from the buffer to be used as a secret key to transmit a message in a slot and then remove those bits. With this scheme we are...

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results...

This thesis focuses on the Shannon capacity of a graph. Suppose we want to send a message across a channel to a receiver. The general question is about the effective size of an alphabet in a model such that the receiver may recover the original message without errors. To answer this question Shannon defined the Shannon capacity of a graph, Θ(G), and stated Shannon’s Theorem. We study an article...