The Inducibility of Graphs on Four Vertices

Hirst, James

doi:10.48550/arxiv.1109.1592

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2013

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Cited by 2 publications

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“…It is crucial to know all forced zero eigenvectors during the rounding step because a small but uncontrolled perturbation of Q τ i may result in negative eigenvalues. Flagmatic 2.0 takes care of this by ensuring that the column space of the matrix R in (11) is orthogonal to all forced zero eigenvectors of Q τ i (when an extremal construction is supplied using the function set_extremal_construction).…”

Section: Proof Certificatesmentioning

confidence: 99%

Minimum Number ofk-Cliques in Graphs with Bounded Independence Number

Pikhurko¹,

Vaughan²

2013

Combinator. Probab. Comp.

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Section: Proof Certificatesmentioning

confidence: 99%

Minimum Number ofk-Cliques in Graphs with Bounded Independence Number

Pikhurko¹,

Vaughan²

2013

Combinator. Probab. Comp.

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“…The inducibility of a graph H is defined as lim n→∞ max G d(H; G), where the maximum is over all n-vertex graphs G. This natural parameter has been investigated for several types of graphs H. E.g., complete bipartite and multipartite graphs [1,2], very small graphs [7,13] and blow-up graphs [11].…”

Section: Introductionmentioning

confidence: 99%

On the 3‐Local Profiles of Graphs

Huang

Linial

Naves³

et al. 2013

Journal of Graph Theory

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For a graph G, let pi(G),i=0,...,3 be the probability that three distinct random vertices span exactly i edges. We call (p0(G),...,p3(G)) the 3‐local profile of G. We investigate the set scriptS3⊂double-struckR4 of all vectors (p0,...,p3) that are arbitrarily close to the 3‐local profiles of arbitrarily large graphs. We give a full description of the projection of S3 to the (p0,p3) plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality p0+p3≥14. We also give a full description of the triangle‐free case, i.e. the intersection of S3 with the hyperplane p3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.

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The Inducibility of Graphs on Four Vertices

Cited by 2 publications

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Minimum Number ofk-Cliques in Graphs with Bounded Independence Number

Minimum Number ofk-Cliques in Graphs with Bounded Independence Number

On the 3‐Local Profiles of Graphs

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