43,935 research outputs found
Dual concepts of almost distance-regularity and the spectral excess theorem
Generally speaking, ‘almost distance-regular’ graphs are graphs that share some, but not necessarily all, regularity properties that characterize distance-regular graphs. In this paper we first
propose two dual concepts of almost distance-regularity. In some cases, they coincide with concepts introduced before by other authors, such as partially distance-regular graphs. Our study focuses on finding out when almost distance-regularity leads to distance-regularity. In particular, some ‘economic’ (in the sense of minimizing the number of conditions) old and new
characterizations of distance-regularity are discussed. Moreover, other characterizations based on the average intersection numbers and the recurrence coefficients are obtained. In some cases, our results can also be seen as a generalization of the
so-called spectral excess theorem for distance-regular graphs.Peer Reviewe
When almost distance-regularity attains distance-regularity
Generally speaking, `almost distance-regular graphs' are graphs which share some, but
not necessarily all, regularity properties that characterize distance-regular graphs. In
this paper we rst propose four basic di erent (but closely related) concepts of almost
distance-regularity. In some cases, they coincide with concepts introduced before by
other authors, such as walk-regular graphs and partially distance-regular graphs. Here
it is always assumed that the diameter D of the graph attains its maximum possible
value allowed by its number d+1 of di erent eigenvalues; that is, D = d, as happens in
every distance-regular graph. Our study focuses on nding out when almost distance-
regularity leads to distance-regularity. In other words, some `economic' (in the sense
of minimizing the number of conditions) old and new characterizations of distance-
regularity are discussed. For instance, if A0;A1; : : : ;AD and E0;E1; : : : ;Ed denote,
respectively, the distance matrices and the idempotents of the graph; and D and A
stand for their respective linear spans, any of the two following `dual' conditions su ce:
(a) A0;A1;AD 2 A; (b) E0;E1;Ed 2 D. Moreover, other characterizations based on
the preintersection parameters, the average intersection numbers and the recurrence
coe cients are obtained. In some cases, our results can be also seen as a generalization
of the so-called spectral excess theorem for distance-regular graphs.Postprint (published version
On distance-regularity in graphs
AbstractIf A is the adjacency matrix of a graph G, then Ai is the adjacency matrix of the graph on the same vertex set in which a pair of vertices is adjacent if and only if their distance apart is i in G. If G is distance-regular, then Ai is a polynomial of degree i in A. It is shown that the converse is also true. If Ai is a polynomial in A, not necessarily of degree i, G is said to be distance-polynomial. It is shown that this is a larger class of graphs and some of its properties are investigated
Distance regularity in buildings and structure constants in Hecke algebras
In this paper we define generalised spheres in buildings using the simplicial
structure and Weyl distance in the building, and we derive an explicit formula
for the cardinality of these spheres. We prove a generalised notion of distance
regularity in buildings, and develop a combinatorial formula for the
cardinalities of intersections of generalised spheres. Motivated by the
classical study of algebras associated to distance regular graphs we
investigate the algebras and modules of Hecke operators arising from our
generalised distance regularity, and prove isomorphisms between these algebras
and more well known parabolic Hecke algebras. We conclude with applications of
our main results to non-negativity of structure constants in parabolic Hecke
algebras, commutativity of algebras of Hecke operators, double coset
combinatorics in groups with -pairs, and random walks on the simplices of
buildings.Comment: J. Algebra, to appea
Cut distance identifying graphon parameters over weak* limits
The theory of graphons comes with the so-called cut norm and the derived cut
distance. The cut norm is finer than the weak* topology. Dole\v{z}al and
Hladk\'y [Cut-norm and entropy minimization over weak* limits, J. Combin.
Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a
cut distance accumulation graphon can be pinpointed in the set of weak*
accumulation points as a minimizer of the entropy. Motivated by this, we study
graphon parameters with the property that their minimizers or maximizers
identify cut distance accumulation points over the set of weak* accumulation
points. We call such parameters cut distance identifying. Of particular
importance are cut distance identifying parameters coming from subgraph
densities, t(H,*). This concept is closely related to the emerging field of
graph norms, and the notions of the step Sidorenko property and the step
forcing property introduced by Kr\'al, Martins, Pach and Wrochna [The step
Sidorenko property and non-norming edge-transitive graphs, J. Combin. Theory
Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if
and only if it is step Sidorenko, and that if a graph is norming then it is
step forcing. Further, we study convexity properties of cut distance
identifying graphon parameters, and find a way to identify cut distance limits
using spectra of graphons. We also show that continuous cut distance
identifying graphon parameters have the "pumping property", and thus can be
used in the proof of the the Frieze-Kannan regularity lemma.Comment: 48 pages, 3 figures. Correction when treating disconnected norming
graphs, and a new section 3.2 on index pumping in the regularity lemm
Some families of orthogonal polynomials of a discrete variable and their applications to graphs and codes
We present some related families of orthogonal polynomials of a discrete variable
and survey some of their applications in the study of (distance-regular) graphs and
(completely regular) codes. One of the main peculiarities of such orthogonal systems
is their non-standard normalization condition, requiring that the square norm of each
polynomial must equal its value at a given point of the mesh. For instance, when they
are de¯ned from the spectrum of a graph, one of these families is the system of the pre-
distance polinomials which, in the case of distance-regular graphs, turns out to be the
sequence of distance polinomials. The applications range from (quasi-spectral) char-
acterizations of distance-regular graphs, walk-regular graphs, local distance-regularity
and completely regular codes, to some results on representation theory
A version of Szemer\'edi's regularity lemma for multicolored graphs and directed graphs that is suitable for induced graphs
In this manuscript we develop a version of Szemer\'edi's regularity lemma
that is suitable for analyzing multicolorings of complete graphs and directed
graphs. In this, we follow the proof of Alon, Fischer, Krivelevich and M.
Szegedy [Combinatorica, 20(4) (2000), 451--476] who prove a similar result for
graphs.
The purpose is to extend classical results on dense hereditary properties,
such as the speed of the property or edit distance, to the above-mentioned
combinatorial objects.Comment: 11 page
Stability and Error Analysis for Optimization and Generalized Equations
Stability and error analysis remain challenging for problems that lack
regularity properties near solutions, are subject to large perturbations, and
might be infinite dimensional. We consider nonconvex optimization and
generalized equations defined on metric spaces and develop bounds on solution
errors using the truncated Hausdorff distance applied to graphs and epigraphs
of the underlying set-valued mappings and functions. In the process, we extend
the calculus of such distances to cover compositions and other constructions
that arise in nonconvex problems. The results are applied to constrained
problems with feasible sets that might have empty interiors, solution of KKT
systems, and optimality conditions for difference-of-convex functions and
composite functions
Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform
We enumerate the inequivalent self-dual additive codes over GF(4) of
blocklength n, thereby extending the sequence A090899 in The On-Line
Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a
well-known interpretation as quantum codes. They can also be represented by
graphs, where a simple graph operation generates the orbits of equivalent
codes. We highlight the regularity and structure of some graphs that correspond
to codes with high distance. The codes can also be interpreted as quadratic
Boolean functions, where inequivalence takes on a spectral meaning. In this
context we define PAR_IHN, peak-to-average power ratio with respect to the
{I,H,N}^n transform set. We prove that PAR_IHN of a Boolean function is
equivalent to the the size of the maximum independent set over the associated
orbit of graphs. Finally we propose a construction technique to generate
Boolean functions with low PAR_IHN and algebraic degree higher than 2.Comment: Presented at Sequences and Their Applications, SETA'04, Seoul, South
Korea, October 2004. 17 pages, 10 figure
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