Lifts Of Graphs Research Articles

Distance and adjacency spectra and eigenspaces for three (di)graph lifts: A unified approach

By the induced character theory, we first establish the irreducible decomposition of a permutation representation. Then, using it as a unified tool, we derive a decomposition formula for the distance spectrum and eigenspace of a regular lift of a graph, the adjacency spectra and eigenspaces of a relative lift and a ramified uniform lift of a digraph. The latter results are a complement of Dalfó et al. (2021) [6] and Deng et al. (2007) [8].

Voltage Lifts of Graphs from a Category Theory Viewpoint

ABSTRACT We prove that the notion of a voltage graph lift comes from an adjunction between the category of voltage graphs and the category of group labeled graphs.

A Unifying Framework to Construct QC-LDPC Tanner Graphs of Desired Girth

This paper presents a unifying framework to construct low-density parity-check (LDPC) codes with associated Tanner graphs of desired girth. Towards this goal, we highlight the role that a certain square matrix that appears in the product of the parity-check matrix with its transpose has in the construction of codes with graphs of desired girth and further explore it in order to generate the set of necessary and sufficient conditions for a Tanner graph to have a given girth between 6 and 12. For each such girth, we present algorithms to construct codes of the desired girth and we show how to use them to compute the minimum necessary value of the lifting factor. For girth larger than 12, we show how to use multi-step graph lifting methods to deterministically modify codes in order to increase their girth. We also give a new perspective on LDPC protograph-based parity-check matrices by viewing them as rows of a parity-check matrix equal to the sum of certain permutation matrices and obtain an important connection between all protographs and those with variable nodes of degree 2. We also show that the results and methodology that we develop for the all-one protograph can be used and adapted to analyze the girth of the Tanner graph of any parity-check matrix and demonstrate how this can be done using a well-known irregular, multi-edge protograph specified by the NASA Consultative Committee for Space Data Systems (CCSDS). Throughout the paper, we exemplify our theoretical results with constructions of LDPC codes with Tanner graphs of any girth between 6 and 14 and give sufficient conditions for a multi-step lifted parity-check matrix to have girth between 14 and 22.

On the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs

The universal adjacency matrix U of a graph Γ, with adjacency matrix A, is a linear combination of A, the diagonal matrix D of vertex degrees, the identity matrix I, and the all-1 matrix J with real coefficients, that is, , with and . Thus, in particular cases, U may be the adjacency matrix, the Laplacian, the signless Laplacian, and the Seidel matrix. In this paper, we develop a method for determining the universal spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular or not). As an example, the method is applied to give an efficient algorithm to determine the characteristic polynomial of the Laplacian matrix of the symmetric squares of odd cycles, together with closed formulas for some of their eigenvalues.

Pre-Demosaic Graph-Based Light Field Image Compression

An unfocused plenoptic light field (LF) camera places an array of microlenses in front of an image sensor in order to separately capture different directional rays arriving at an image pixel. Using a conventional Bayer pattern, data captured at each pixel is a single color component (R, G or B).The sensed data then undergoes demosaicking (interpolation of RGB components per pixel) and conversion to an array of sub-aperture images (SAIs). In this paper, we propose a new LF image coding scheme based on graph lifting transform (GLT), where the acquired sensor data are coded in the original captured form without pre-processing. Specifically, we directly map raw sensed color data to the SAIs, resulting in sparsely distributed color pixels on 2D grids, and perform demosaicking at the receiver after decoding. To exploit spatial correlation among the sparse pixels, we propose a novel intra-prediction scheme, where the prediction kernel is determined according to the local gradient estimated from already coded neighboring pixel blocks. We then connect the pixels by forming a graph, modeling the prediction residuals statistically as a Gaussian Markov Random Field (GMRF). The optimal edge weights are computed via a graph learning method using a set of training SAIs. The residual data is encoded via low-complexity GLT. Experiments show that at high PSNRs -- important for archiving and instant storage scenarios -- our method outperformed significantly a conventional light field image coding scheme with demosaicking followed by High Efficiency Video Coding (HEVC).

Spectra and eigenspaces of arbitrary lifts of graphs

We describe, in a very explicit way, a method for determining the spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular or not).

Designing Protograph-Based Quasi-Cyclic Spatially Coupled LDPC Codes With Large Girth

Spatially coupled (SC) low-density parity-check (LDPC) codes can achieve capacity approaching performance with low message recovery latency when using sliding window (SW) decoding. An SC-LDPC code constructed from a protograph can be generated by first coupling a chain of block protographs and then lifting the coupled protograph using permutation matrices. In this paper, we introduce a systematic design to eliminate 4-cycles in a coupled protograph. Further using a quasi-cyclic (QC) lifting, we introduce a procedure for constructing QC-SC-LDPC codes of girth at least eight. This can be interpreted as a multi-stage graph lifting process that yields a greater flexibility in designing QC-SC-LDPC codes with a large girth than previous approaches. Simulation results show the design leads to improved decoding performance, particularly in the error floor, compared to random constructions. Finally, we determine the minimum coupling width required to eliminate 4-cycles in a coupled protograph.

A new proof of Friedman's second eigenvalue theorem and its extension to random lifts

It was conjectured by Alon and proved by Friedman that a random $d$-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most $2\sqrt{d-1} +o(1)$ with probability tending to one as the size of the graph tends to infinity. We give a new proof of this statement. We also study related questions on random $n$-lifts of graphs and improve a recent result by Friedman and Kohler.

Compression of Hyperspectral Scenes through Integer-to-Integer Spectral Graph Transforms

Hyperspectral images are depictions of scenes represented across many bands of the electromagnetic spectrum. The large size of these images as well as their unique structure requires the need for specialized data compression algorithms. The redundancies found between consecutive spectral components and within components themselves favor algorithms that exploit their particular structure. One novel technique with applications to hyperspectral compression is the use of spectral graph filterbanks such as the GraphBior transform, that leads to competitive results. Such existing graph based filterbank transforms do not yield integer coefficients, making them appropriate only for lossy image compression schemes. We propose here two integer-to-integer transforms that are used in the biorthogonal graph filterbanks for the purpose of the lossless compression of hyperspectral scenes. Firstly, by applying a Triangular Elementary Rectangular Matrix decomposition on GraphBior filters and secondly by adding rounding operations to the spectral graph lifting filters. We examine the merit of our contribution by testing its performance as a spatial transform on a corpus of hyperspectral images; and share our findings through a report and analysis of our results.

$\delta$-Connectivity in Random Lifts of Graphs

Amit and Linial have shown that a random lift of a connected graph with minimum degree $\delta\ge3$ is asymptotically almost surely (a.a.s.) $\delta$-connected and mentioned the problem of estimating this probability as a function of the degree of the lift. Using a connection between a random $n$-lift of a graph and a randomly generated subgroup of the symmetric group on $n$-elements, we show that this probability is at least $1 - O\left(\frac{1}{n^{\gamma(\delta)}}\right)$ where $\gamma(\delta)>0$ for $\delta\ge 5$ and it is strictly increasing with $\delta$. We extend this to show that one may allow $\delta$ to grow slowly as a function of the degree of the lift and the number of vertices and still obtain that random lifts are a.a.s. $\delta$-connected. We also simplify a later result showing a lower bound on the edge expansion of random lifts. On a related note, we calculate the probability that a subgroup of a wreath product of symmetric groups generated by random generators is transitive, extending a well known result of Dixon which covers the case for subgroups of the symmetric group.

Multiresolution analysis over graphs for a motor imagery based online BCI game

Multiresolution analysis (MRA) over graph representation of EEG data has proved to be a promising method for offline brain–computer interfacing (BCI) data analysis. For the first time we aim to prove the feasibility of the graph lifting transform in an online BCI system. Instead of developing a pointer device or a wheel-chair controller as test bed for human–machine interaction, we have designed and developed an engaging game which can be controlled by means of imaginary limb movements. Some modifications to the existing MRA analysis over graphs for BCI have also been proposed, such as the use of common spatial patterns for feature extraction at the different levels of decomposition, and sequential floating forward search as a best basis selection technique. In the online game experiment we obtained for three classes an average classification rate of 63.0% for fourteen naive subjects. The application of a best basis selection method helps significantly decrease the computing resources needed. The present study allows us to further understand and assess the benefits of the use of tailored wavelet analysis for processing motor imagery data and contributes to the further development of BCI for gaming purposes.

Hamilton cycles in random lifts of graphs

For a graph G the random n-lift of G is obtained by replacing each of its vertices by a set of n vertices, and joining a pair of sets by a random matching whenever the corresponding vertices of G are adjacent. We show that asymptotically almost surely the random lift of a graph G is Hamiltonian, provided G has the minimum degree at least 5 and contains two disjoint Hamiltonian cycles whose union is not a bipartite graph.

Random Lifts of Graphs are Highly Connected

In this note we study asymptotic properties of random lifts of graphs introduced by Amit and Linial as a new model of random graphs. Given a base graph $G$ and an integer $n$, a random lift of $G$ is obtained by replacing each vertex of $G$ by a set of $n$ vertices, and joining these sets by random matchings whenever the corresponding vertices of $G$ are adjacent. In this paper we study connectivity properties of random lifts. We show that the size of the largest topological clique in typical random lifts, with $G$ fixed and $n\rightarrow\infty$, is equal to the maximum degree of the core of $G$ plus one. A similar idea can be used to prove that for any graph $G$ with $\delta(G)\geq2k-1$ almost every random lift of $G$ is $k$-linked.

Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on non-commuting permutation matrices. We also show encoder and decoder implementations for these codes.

Tight products and graph expansion

AbstractIn this article, we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G1 and G2, if V(H) = V(G1) × V(G2) and both projection maps V(H)→V(G1) and V(H)→V(G2) are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is NP‐hard to decide). We investigate the conditions under which this is possible. This perspective yields a new characterization of class‐1 (2k+ 1)‐regular graphs. We also obtain a new model of random d‐regular graphs whose second eigenvalue is almost surely at most O(d3/4). This construction resembles random graph lifts, but requires fewer random bits. © 2011 Wiley Periodicals, Inc. J Graph Theory

On the girth of voltage graph lifts

We study the potential and limitations of the voltage graph construction for producing small regular graphs of large girth. We determine the relation between the girth of the base graph and the lift, and we show that any base graph can be lifted to a graph of arbitrarily large girth. We determine upper bounds on the girths of voltage graphs with respect to the nilpotency class in the case of nilpotent groups or the length of the derived series in the case of solvable voltage groups. These results suggest the use of perfect groups, which we use to construct the smallest known cubic graphs of girths 29 and 30. We also construct the smallest known (5, 10)-graphs and (7, 8)-graphs.

Word maps and spectra of random graph lifts

We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ1 and ρ are the spectral radii of G and T respectively, then, as shown by Friedman (Graphs Duke Math J 118 (2003), 19–35), in almost every n-lift H of G, all “new” eigenvalues of H are ≤ O(λ ρ1/2). Here we improve this bound to O(λ ρ2/3). It is conjectured in (Friedman, Graphs Duke Math J 118 (2003) 19–35) that the statement holds with the bound ρ + o(1) which, if true, is tight by (Greenberg, PhD thesis, 1995). For G a bouquet with d/2 loops, our arguments yield a simple proof that almost every d-regular graph has second eigenvalue O(d2/3). For the bouquet, Friedman (2008). has famously proved the (nearly?) optimal bound of . Central to our work is a new analysis of formal words. Let w be a formal word in letters g,…,g. The word map associated with w maps the permutations σ1,…,σk ∈ Sn to the permutation obtained by replacing for each i, every occurrence of gi in w by σi. We investigate the random variable X that counts the fixed points in this permutation when the σi are selected uniformly at random. The analysis of the expectation 𝔼(X) suggests a categorization of formal words which considerably extends the dichotomy of primitive vs. imprimitive words. A major ingredient of a our work is a second categorization of formal words with the same property. We establish some results and make a few conjectures about the relation between the two categorizations. These conjectures suggest a possible approach to (a slightly weaker version of) Friedman's conjecture. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica (Nica, Random Struct Algorithms 5 (1994), 703–730), which determines, for every fixed w, the limit distribution (as n →∞) of X. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w = ud for some word u. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

The chromatic number of random lifts of

Abstract Amit, Linial, and Matousek (Random lifts of graphs III: independence and chromatic number, Random Struct. Algorithms, 2002) have raised the following question: Is the chromatic number of random h-lifts of K 5 asymptotically (for h → ∞ ) almost surely (a.a.s.) equal to a single number? In this paper, we offer the following partial result: The chromatic number of a random lift of K 5 e is a.a.s. three. We actually prove a stronger statement where K 5 e can be replaced by a graph obtained from joining a cycle to a stable set.

Hamilton cycles in random lifts of graphs

An n - lift of a graph K is a graph with vertex set V ( K ) × [ n ] , and for each edge ( i , j ) ∈ E ( K ) there is a perfect matching between { i } × [ n ] and { j } × [ n ] . If these matchings are chosen independently and uniformly at random then we say that we have a random n -lift. We show that there are constants h 1 , h 2 such that if h ≥ h 1 then a random n -lift of the complete graph K h is hamiltonian whp and if h ≥ h 2 then a random n -lift of the complete bipartite graph K h , h is hamiltonian whp .

Random Lifts of Graphs: Edge Expansion

We continue the study of random lifts of graphs initiated in [4]. Here we study the possibility of generating graphs with high edge expansion as random lifts. Along the way, we introduce the method of -nets into the study of random structures. This enables us to improve (slightly) the known bounds for the edge expansion of regular graphs.