The zero-error capacity of a classical channel is expressed in terms of the independence number of some graph and its tensor powers. This quantity is hard to compute even for small graphs such as the cycle of length seven, so upper bounds such as the Lovász theta function play an important role in zero-error communication. In this paper, we show that the Lovász theta function is an upper bound ...

The Lovász theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovász theta number toward the chromatic number of G, which is shown to be equal to the fractional ch...

A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order n ≥ 12 and size at least ⌊ 11n−18 2 ⌋ contains three disjoint theta graphs. As a corollary, every graph of order n ≥ 12 and size at least ⌊ 11n−18 2 ⌋ contains three disjoint cycles of even length. 1. Terminology and introduction In this paper, we only co...

We observe that ω(G) + χ(S( G)) = n = ω(S( G)) + χ(G) for any graph G with n vertices, where G is any acyclic orientation of G and where S( G) is the (complement of the) auxiliary line graph introduced in [1]. (Where as usual, ω and χ denote the clique number and the chromatic number.) It follows that, for any graph parameter β(G) sandwiched between ω(G) and χ(G), then Φβ( G) := n−β(S( G)) is s...

The diamond is the graph obtained by removing an edge from complete on 4 vertices. A (\(P_6\), diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or diamond. In this paper we show that chromatic number of G larger than maximum 6 and clique G. We do reducing problem imperfect graphs via Strong Perfect Graph Theorem, dividing into several cases, giving proper colouri...

We show that the second homology of configuration spaces a planar graph is generated under operations embedding, disjoint union, and edge stabilization by three atomic graphs: cycle with one edge, star edges, theta four edges. give an example non-planar for which this statement false.

Disruptions in functional connectivity and dysfunctional brain networks are considered to be a neurological hallmark of neurodevelopmental disorders. Despite the vast literature on functional brain connectivity in typical brain development, surprisingly few attempts have been made to characterize brain network integrity in neurodevelopmental disorders. Here we used resting-state EEG to characte...

BACKGROUND Functional magnetic resonance imaging (fMRI) resting-state studies show generalized social anxiety disorder (gSAD) is associated with disturbances in networks involved in emotion regulation, emotion processing, and perceptual functions, suggesting a network framework is integral to elucidating the pathophysiology of gSAD. However, fMRI does not measure the fast dynamic interconnectio...

A homomorphism from a graph X to a graph Y is an adjacency preserving mapping f : V (X)→ V (Y ). We consider a nonlocal game in which Alice and Bob are trying to convince a verifier with certainty that a graph X admits a homomorphism to Y . This is a generalization of the well-studied graph coloring game. Via systematic study of quantum homomorphisms we prove new results for graph coloring. Mos...