28 research outputs found
The spectral excess theorem for distance-biregular graphs
The spectral excess theorem for distance-regular graphs states that a regular
(connected) graph is distance-regular if and only if its spectral-excess equals its
average excess. A bipartite graphPeer ReviewedPostprint (published version
A general method to obtain the spectrum and local spectra of a graph from its regular partitions
It is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, a method that gives all the spectrum, and also the local spectra, of a graph from the quotient matrices of some of its regular partitions, is proposed. Moreover, from such partitions, the C-local multiplicities of any class of vertices C is also determined, and some applications of these parameters in the characterization of completely regular codes and their inner distributions are described. As examples, it is shown how to find the eigenvalues and (local) multiplicities of walk-regular, distance-regular, and distance-biregular graphs.Partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Governmentunder projects PGC2018-095471-B-I00 and MTM2017-83271-R. Also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement no. 73492
Algebraic characterizations of bipartite distance-regular graphs
Bipartite graphs are combinatorial objects bearing some interesting symmetries. Thus, their spectra—eigenvalues of its adjacency
matrix—are symmetric about zero, as the corresponding eigenvectors come into pairs. Moreover, vertices in the same (respectively, different) independent set are always at even (respectively, odd) distance. Both properties have well-known consequences in most properties and parameters of such graphs.
Roughly speaking, we could say that the conditions for a given property to hold in a general graph can be somehow relaxed to guaranty the same property for a bipartite graph. In this paper we comment upon this phenomenon in the framework of distance-regular graphs for which several characterizations, both of combinatorial or algebraic nature, are known. Thus, the
presented characterizations of bipartite distance-regular graphs involve such parameters as the numbers of walks between vertices (entries of the powers of the adjacency matrix A), the crossed local multiplicities (entries of the idempotents or eigenprojectors), the predistance polynomials, etc. For instance, it is known that a graph G, with eigenvalues > > · · · > and diameter D = d, is distance-regular if and only if its idempotents and belong to the vector space D spanned by its distance matrices I,A,, . . .. In contrast with this, for the same result to be true in the case of bipartite graphs, only ∈ D need to be required.Peer Reviewe
A characterization and an application of weight-regular partitions of graphs
A natural generalization of a regular (or equitable) partition of a graph,
which makes sense also for non-regular graphs, is the so-called weight-regular
partition, which gives to each vertex a weight that equals the
corresponding entry of the Perron eigenvector . This
paper contains three main results related to weight-regular partitions of a
graph. The first is a characterization of weight-regular partitions in terms of
double stochastic matrices. Inspired by a characterization of regular graphs by
Hoffman, we also provide a new characterization of weight-regularity by using a
Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular
graphs. In addition, we show an application of weight-regular partitions to
study graphs that attain equality in the classical Hoffman's lower bound for
the chromatic number of a graph, and we show that weight-regularity provides a
condition under which Hoffman's bound can be improved
Explicit two-sided unique-neighbor expanders
We study the problem of constructing explicit sparse graphs that exhibit
strong vertex expansion. Our main result is the first two-sided construction of
imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets
contained in both the left and right bipartitions exhibit unique-neighbor
expansion, along with algebraic properties relevant to constructing quantum
codes.
Our constructions are obtained from instantiations of the tripartite line
product of a large tripartite spectral expander and a sufficiently good
constant-sized unique-neighbor expander, a new graph product we defined that
generalizes the line product in the work of Alon and Capalbo and the routed
product in the work of Asherov and Dinur.
To analyze the vertex expansion of graphs arising from the tripartite line
product, we develop a sharp characterization of subgraphs that can arise in
bipartite spectral expanders, generalizing results of Kahale, which may be of
independent interest.
By picking appropriate graphs to apply our product to, we give a strongly
explicit construction of an infinite family of -biregular graphs
(for large enough and ) where all sets with
fewer than a small constant fraction of vertices have
unique-neighbors (assuming ).
Additionally, we can also guarantee that subsets of vertices of size up to
expand losslessly.Comment: New version contains stronger result, and many new technical
ingredients. 45 pages, 2 figure
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Distance-Biregular Graphs and Orthogonal Polynomials
This thesis is about distance-biregular graphs– when they exist, what algebraic and structural properties they have, and how they arise in extremal problems.
We develop a set of necessary conditions for a distance-biregular graph to exist. Using these conditions and a computer, we develop tables of possible parameter sets for distancebiregular graphs. We extend results of Fiol, Garriga, and Yebra characterizing distance-regular graphs to characterizations of distance-biregular graphs, and highlight some new
results using these characterizations. We also extend the spectral Moore bounds of Cioaba et al. to semiregular bipartite graphs, and show that distance-biregular graphs arise as extremal examples of graphs meeting the spectral Moore bound
Edge-distance-regular graphs
Edge-distance-regularity is a concept recently introduced by the authors which is
similar to that of distance-regularity, but now the graph is seen from each of its edges
instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edge-distance-regular when it is distance-regular around each of its edges and with
the same intersection numbers for any edge taken as a root. In this paper we study
this concept, give some of its properties, such as the regularity of Γ, and derive some
characterizations. In particular, it is shown that a graph is edge-distance-regular if and only if its k-incidence matrix is a polynomial of degree k in A multiplied by the
(standard) incidence matrix. Also, the analogue of the spectral excess theorem for
distance-regular graphs is proved, so giving a quasi-spectral characterization of edgedistance-regularity. Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.Preprin