63 research outputs found
Distinguishing graphs by their left and right homomorphism profiles
We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte
uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu.
We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are
specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and
Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from
multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and
Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky.
Unifying the study of these and related problems is the notion of the left and right homomorphism profiles
of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164
The equivariant topology of stable Kneser graphs
The stable Kneser graph , , , introduced by Schrijver
\cite{schrijver}, is a vertex critical graph with chromatic number , its
vertices are certain subsets of a set of cardinality . Bj\"orner and de
Longueville \cite{anders-mark} have shown that its box complex is homotopy
equivalent to a sphere, \Hom(K_2,SG_{n,k})\homot\Sphere^k. The dihedral group
acts canonically on , the group with 2 elements acts
on . We almost determine the -homotopy type of
\Hom(K_2,SG_{n,k}) and use this to prove the following results. The graphs
are homotopy test graphs, i.e. for every graph and such
that \Hom(SG_{2s,4},H) is -connected, the chromatic number
is at least . If and then
is not a homotopy test graph, i.e.\ there are a graph and an such
that \Hom(SG_{n,k}, G) is -connected and .Comment: 34 pp
Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors
Two graphs and are homomorphism indistinguishable over a class of
graphs if for all graphs the number of
homomorphisms from to is equal to the number of homomorphisms from
to . Many natural equivalence relations comparing graphs such as (quantum)
isomorphism, spectral, and logical equivalences can be characterised as
homomorphism indistinguishability relations over certain graph classes.
Abstracting from the wealth of such instances, we show in this paper that
equivalences w.r.t. any self-complementarity logic admitting a characterisation
as homomorphism indistinguishability relation can be characterised by
homomorphism indistinguishability over a minor-closed graph class.
Self-complementarity is a mild property satisfied by most well-studied logics.
This result follows from a correspondence between closure properties of a graph
class and preservation properties of its homomorphism indistinguishability
relation.
Furthermore, we classify all graph classes which are in a sense finite
(essentially profinite) and satisfy the maximality condition of being
homomorphism distinguishing closed, i.e. adding any graph to the class strictly
refines its homomorphism indistinguishability relation. Thereby, we answer
various question raised by Roberson (2022) on general properties of the
homomorphism distinguishing closure.Comment: 26 pages, 1 figure, 1 tabl
Measures on the square as sparse graph limits
We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional "factors". We show that limits of convergent graph sequences in this metric can be represented by symmetric Borel measures on [0, 1](2). This leads to a generalization of dense graph limit theory to sparse graph sequences. (C) 2019 Elsevier Inc. All rights reserved
Upper tails and independence polynomials in random graphs
The upper tail problem in the Erd\H{o}s--R\'enyi random graph
asks to estimate the probability that the number of
copies of a graph in exceeds its expectation by a factor .
Chatterjee and Dembo showed that in the sparse regime of as
with for an explicit ,
this problem reduces to a natural variational problem on weighted graphs, which
was thereafter asymptotically solved by two of the authors in the case where
is a clique. Here we extend the latter work to any fixed graph and
determine a function such that, for as above and any fixed
, the upper tail probability is , where is the maximum degree of . As it turns out, the
leading order constant in the large deviation rate function, , is
governed by the independence polynomial of , defined as where is the number of independent sets of size in . For
instance, if is a regular graph on vertices, then is the
minimum between and the unique positive solution of
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