M. Laczkovich, A. Máthé · Open MIND · 2026
We say that a plane set $A$ is {\it graph-null,} if there is a function $g\colon [0,1] \to \mathbb{R}$ such that $λ_2 (A+{\rm graph}\, g)=0$. A plane set $A$ has the {\it translational Kakeya property} if, for every translated copy $A'$ of $A$ and for every $ε>0$, there is a finite sequence of vertical and horizontal translations bringing $A$ to $A'$ such that the area touched during the horizontal translations is less than $ε$. These properties are equivalent if $A$ is compact. We show that the graph of every absolutely continuous function is graph-null. Also, the graph of a typical continuous function is graph-null. Therefore, there are nowhere differentiable continuous functions whose graphs are graph-null. Still, we show that there exists a continuous function whose graph is not graph-null.
Cited by 0Open access
Laczkovich, M., Máthé, A. · arXiv (Cornell University) · 2026
Cited by 0Open access
M. Laczkovich, A. Máthé · arXiv (Cornell University) · 2026
We say that a plane set $A$ is {\it graph-null,} if there is a function $g\colon [0,1] \to \mathbb{R}$ such that $λ_2 (A+{\rm graph}\, g)=0$. A plane set $A$ has the {\it translational Kakeya property} if, for every translated copy $A'$ of $A$ and for every $ε>0$, there is a finite sequence of vertical and horizontal translations bringing $A$ to $A'$ such that the area touched during the horizontal translations is less than $ε$. These properties are equivalent if $A$ is compact. We show that the graph of every absolutely continuous function is graph-null. Also, the graph of a typical continuous function is graph-null. Therefore, there are nowhere differentiable continuous functions whose graphs are graph-null. Still, we show that there exists a continuous function whose graph is not graph-null.
Cited by 0Open access
Heiko Müller (7901), Francesco Mancuso (73553) · Figshare · 2015
<p>A) PubMed IDs (PMIDx) and genes (gx) are represented by vertices of a bipartite graph. An edge indicates that a gene has been reported as differentially regulated in a specific publication. B–D) Occurrence probabilities of the bipartite graph shown in panel A as determined by six different null-models for PMID1 (B), PMID2 (C) and PMID3 (D): GR - generic randomization, GVD - gene vertex degree, LVD - list vertex degree, BN - binomial, HG - hypergeometric, ES - edge-swapping. See text for details of different null-models.</p>
Cited by 1Open access
Jack Garbus, Christopher Brissette, George M. Slota · 2020
The generation of uniformly-random graphs is a key analytic tool for hypothesis testing throughout social network analysis. This work specifically optimizes the generation of large-scale simple uniform random graphs. We consider the separate but related problems of generating such a graph from an existing edge list and the problem of generating a graph from only a degree distribution. To address these problems, we implement an efficient parallel Markov Chain Monte Carlo process for double-edge swapping, a fast and parallel method for edge-skipping edge list generation, and a novel method to solve for valid inputs to our edge-skipping generator. Our double-edge swapping procedure is considerably faster than prior parallel methods, our edge generator uses state-of-the-art methods, and the algorithmic approach we have to solve for valid edge-skipping inputs addresses the often significant shortcomings of current approaches.
Cited by 5
Diari Indriati, Muhammad Yogi Sentosa, Zhafirah Miranti Verdiana et al. · European Journal of Pure and Applied Mathematics · 2025
Given a graph $G(V,E)$ or simply written as $G$. The graph labeling was first introduced in 1960, it is a function that mapping integers or labels to graph elements (vertices, edges, or both of them) which must satisfy some certain criteria. This concepts was applied in some real problems. In 2007, there was a new concept on labeling, i.e., ``vertex irregular total labeling" and ``edge irregular total labeling". In 2017, there was a new concept, i.e., ``vertex irregular reflexive labeling" and ``edge irregular reflexive labeling". The ``edge irregular reflexive $k-$ labeling" of $G$ is a mapping that puts an even number label from $0$ to $2k_v$ to every vertex and a positive integer label from $1$ to $k_e$ to every edge with different weights for each edge. The minimum $k$ of the biggest label among all possible ``edge irregular reflexive $k-$ labeling" of $G$ is called the ``reflexive edge strength" of $G$, denoted as $res(G)$. There are only few result on ``vertex irregular reflexive labeling" or ``edge irregular reflexive labeling" of corona product graphs. Hence, the purpose of this research are to investigate $res$ of corona product of some ladder graphs and null graphs. Therefore, the goal of this study is to examine the $res$ of corona product of some ladder graphs and null graphs. We get the results as follows: $res(SL_n \odot N_m)$ for ``$n \ge 2$ and $m \ge 1$" are ``$\left \lceil \frac{2nm+3n-3}{3} \right \rceil$" for ``$2nm+3n-3 \not \equiv 2,3 \pmod{6}$" and ``$\left \lceil \frac{2nm+3n-3}{3} \right \rceil+1$" for ``$2nm+3n-3 \equiv 2,3 \pmod{6}$". Moreover, $res(DL_n \odot N_m)$ for $n\geq2$ and $m\geq1$ are ``$\left \lceil \frac{2nm+5n-4}{3} \right \rceil$" for ``$2nm+5n-4 \not \equiv 2,3 \pmod{6}$" and ``$\left \lceil \frac{2nm+5n-4}{3} \right \rceil+1$" for ``$2nm+5n-4 \equiv 2,3 \pmod{6}$". These results contribute to developing the theory of reflexive labeling.
Cited by 0Open access
Journal of Mathematical and Computational Science · 2021
A function f: P(G)∪L(G)∪R(G)→C is said to be perfect coloring of the graph G, if f(x)≠f(y) for any two adjoint or incident elements x,y∈P(G)∪L(G)∪R(G). And the PC number χ P (G) is the least colors needed to color a graph by using perfect coloring. In this paper, we prove the results for perfect coloring of corona product(PCCP) of wheel graph family with null, chain, fan and cycle graph, which leads to perfect chromatic number equivalent to ∆+1, where ∆ is the largest degree of the resultant graph in corona product.
Cited by 0Open access
Zeyu Wang, Jinhuan Wang, Yalu Shan et al. · IEEE Transactions on Network Science and Engineering · 2023
Graph classification is an important task widely applied to biochemistry, social networks, and other fields. Since it is a data-dependent problem, insufficient training data will deteriorate the performance of graph classification models. To address this issue, various data augmentation methods have been proposed. However, most existing methods tend to destroy the topological features, leading to a negative impact on information propagation and semantics. While the null model generates new data with the same topological features as the original graph and helps capture the latent information based on topology. Hence, combining the null model with data augmentation for graph classification may be useful in helping models learn graph representations. This paper introduces a novel null model-based augmentation technique for graph classification. Specifically, four standard and four approximate null model-based augmentation methods are developed to verify the effectiveness of the technique. Experimental results on benchmark datasets demonstrate significant performance improvements with the proposed technique. Depending on the design mechanisms of the null models, standard augmentation methods outperform the approximate ones. These findings emphasize the critical role of non-trivial features in enhancing augmentation models for different network structures, providing a new perspective on data augmentation for studying graph classification methods.
Cited by 5
Qi Xuan, Zeyu Wang, Jinhuan Wang et al. · arXiv (Cornell University) · 2021
In network science, the null model is typically used to generate a series of graphs based on randomization as a term of comparison to verify whether a network in question displays some non-trivial features such as community structure. Since such non-trivial features play a significant role in graph classification, the null model could be useful for network data augmentation to enhance classification performance. In this paper, we propose a novel technique that combines the null model with data augmentation for graph classification. Moreover, we propose four standard null model-based augmentation methods and four approximate null model-based augmentation methods to verify and improve the performance of our graph classification technique. Our experiments demonstrate that the proposed augmentation technique has significantly achieved general improvement on the tested datasets. In addition, we find that the standard null model-based augmentation methods always outperform the approximate ones, depending on the design mechanisms of the null models. Our results indicate that the choice of non-trivial features is significant for increasing the performance of augmentation models for different network structures, which also provides a new perspective of data augmentation for studying various graph classification methods.
Cited by 0Open access
Romario Gualdrón-Hurtado, Román Jácome, Rafael S. Suarez et al. · arXiv (Cornell University) · 2026
Inverse problems in imaging are ill-posed, leading to infinitely many solutions consistent with the measurements due to the non-trivial null-space of the sensing matrix. Common image priors promote solutions on the general image manifold, such as sparsity, smoothness, or score function. However, as these priors do not constrain the null-space component, they can bias the reconstruction. Thus, we aim to incorporate meaningful null-space information in the reconstruction framework. Inspired by smooth image representation on graphs, we propose Graph-Smooth Null-Space Representation (GSNR), a mechanism that imposes structure only into the invisible component. Particularly, given a graph Laplacian, we construct a null-restricted Laplacian that encodes similarity between neighboring pixels in the null-space signal, and we design a low-dimensional projection matrix from the $p$-smoothest spectral graph modes (lowest graph frequencies). This approach has strong theoretical and practical implications: i) improved convergence via a null-only graph regularizer, ii) better coverage, how much null-space variance is captured by $p$ modes, and iii) high predictability, how well these modes can be inferred from the measurements. GSNR is incorporated into well-known inverse problem solvers, e.g., PnP, DIP, and diffusion solvers, in four scenarios: image deblurring, compressed sensing, demosaicing, and image super-resolution, providing consistent improvement of up to 4.3 dB over baseline formulations and up to 1 dB compared with end-to-end learned models in terms of PSNR.
Cited by 0Open access
Romario Gualdrón-Hurtado, Román Jácome, Rafael S. Suarez et al. · arXiv (Cornell University) · 2026
Inverse problems in imaging are ill-posed, leading to infinitely many solutions consistent with the measurements due to the non-trivial null-space of the sensing matrix. Common image priors promote solutions on the general image manifold, such as sparsity, smoothness, or score function. However, as these priors do not constrain the null-space component, they can bias the reconstruction. Thus, we aim to incorporate meaningful null-space information in the reconstruction framework. Inspired by smooth image representation on graphs, we propose Graph-Smooth Null-Space Representation (GSNR), a mechanism that imposes structure only into the invisible component. Particularly, given a graph Laplacian, we construct a null-restricted Laplacian that encodes similarity between neighboring pixels in the null-space signal, and we design a low-dimensional projection matrix from the $p$-smoothest spectral graph modes (lowest graph frequencies). This approach has strong theoretical and practical implications: i) improved convergence via a null-only graph regularizer, ii) better coverage, how much null-space variance is captured by $p$ modes, and iii) high predictability, how well these modes can be inferred from the measurements. GSNR is incorporated into well-known inverse problem solvers, e.g., PnP, DIP, and diffusion solvers, in four scenarios: image deblurring, compressed sensing, demosaicing, and image super-resolution, providing consistent improvement of up to 4.3 dB over baseline formulations and up to 1 dB compared with end-to-end learned models in terms of PSNR.
Cited by 0Open access
E S Roberts, T Schlitt · 2011
Tailored graph ensembles as proxies or null models for real networks II: results on directed graphs This article has been downloaded from IOPscience. Please scroll down to see the full text article.
Cited by 0
Mochamad Raffli Ispriyanto, Diari Indriati, Putranto Hadi Utomo · Indonesian Journal of Combinatorics · 2024
Assume <em>G</em> is a graph that is simple, undirected, and connected. If every edge label is a positive integer in the range 1 to <em>k</em><sub>e</sub>, and every vertex label is a non-negative even number from 0 to 2<em>k</em><sub>v</sub>, then a graph <em>G</em> is considered to have an edge irregular reflexive <em>k</em>-labeling, where <em>k</em> is defined as the maximum of <em>k</em><sub>e</sub> and 2<em>k</em><sub>v</sub>. The edge weight <em>w</em><sub>t</sub>(<em>ab</em>) in the graph <em>G</em>, for the labeling <em>λ</em>, is defined as the function <em>w</em><sub>t</sub> applied to the edge <em>ab</em>. The symbol <em>res</em>(<em>G</em>) denotes the reflexive edge strength, which is the largest label of the smallest <em>k</em>. The results of this research are as follows: <em>res</em>(<em>SL</em><sub>m</sub>) for <em>m</em>≥2 is ⌈(3<em>m</em>−3)/3⌉ for 3<em>m</em>−3 ≢ 2, 3 (mod 6), and ⌈(3<em>m</em>−3)/3⌉+1 for 3<em>m</em>−3 ≡ 2, 3 (mod 6). <em>res</em>(<em>Cp</em><sub>n</sub> ⊙ <em>N</em><sub>m</sub>) for <em>n</em>≥2, <em>m</em>≥1 is ⌈(2<em>nm</em>+2<em>n</em>−1)/3⌉ for 2<em>nm</em>+2<em>n</em>−1 ≢ 2, 3 (mod 6), and ⌈(2<em>nm</em>+2<em>n</em>−1)/3⌉+1 for 2<em>nm</em>+2<em>n</em>−1 ≡ 2, 3 (mod 6).
Cited by 0Open access
Gohdar H. Mohiaddin, Khidir R. Sharaf · Science Journal of University of Zakho · 2019
In geographic, the eigenvalues and eigenvectors of transportation network provides many informations about its connectedness. It is proven that the more highly connected in a transportation network G has largest eigenvalue and hence more multiple occurrences of the eigenvalue -1. For a graph G with adjacency matrix A, the multiplicity of the eigenvalue -1 equals the dimension of the null space of the matrix A + I. In this paper, we constructed a high closed zero sum weighting of G and by which its proved that, the dimension of the null space of the eigenvalue -1 is the same as the number of independent variables used in a non-trivial high closed zero sum weighting of the graph. Multiplicity of -1 as an eigenvalue of known graphs and of corona product of certain classes of graphs are determined and two classes of -1- nut graphs are constructed.
Cited by 0Open access
Dalmasso, Cyril, Bar-Hen, Avner, Broët, Philippe · Figshare · 2011
<b>Copyright information:</b>Taken from "A constrained polynomial regression procedure for estimating the local False Discovery Rate"http://www.biomedcentral.com/1471-2105/8/229BMC Bioinformatics 2007;8():229-229.Published online 29 Jun 2007PMCID:PMC1940264.
Cited by 0Open access
Hang Yin, Zipeng Liu, Xiaoyong Peng et al. · ArXiv.org · 2025
Graph unlearning is tailored for GNNs to handle widespread and various graph structure unlearning requests, which remain largely unexplored. The GIF (graph influence function) achieves validity under partial edge unlearning, but faces challenges in dealing with more disturbing node unlearning. To avoid the overhead of retraining and realize the model utility of unlearning, we proposed a novel node unlearning method to reverse the process of aggregation in GNN by embedding reconstruction and to adopt Range-Null Space Decomposition for the nodes' interaction learning. Experimental results on multiple representative datasets demonstrate the SOTA performance of our proposed approach.
Cited by 0Open access
Dalmasso Cyril, Bar-Hen Avner, Broët Philippe · Figshare · 2011
<b>Copyright information:</b>Taken from "A constrained polynomial regression procedure for estimating the local False Discovery Rate"http://www.biomedcentral.com/1471-2105/8/229BMC Bioinformatics 2007;8():229-229.Published online 29 Jun 2007PMCID:PMC1940264.
Cited by 0Open access
François Bienvenu, Florence Débarre, Amaury Lambert · Stochastic Processes and their Applications · 2018
Cited by 7Open access
Irfan Setiawan, Diari Indriati · Indonesian Journal of Combinatorics · 2021
<p>Let <em>G</em>(<em>V</em>,<em>E</em>) be a simple and connected graph which set of vertices is <em>V</em> and set of edges is <em>E</em>. Irregular reflexive <em>k</em>-labeling f on <em>G</em>(<em>V</em>,<em>E</em>) is assignment that carries the numbers of integer to elements of graph, such that the positive integer {1,2, 3,...,<em>k</em><sub>e</sub>} assignment to edges of graph and the even positive integer {0,2,4,...,2<em>k</em><sub>v</sub>} assignment to vertices of graph. Then, we called as edge irregular reflexive <em>k</em>-labelling if every edges has different weight with <em>k</em> = max{<em>k</em><sub>e</sub>,2<em>k</em><sub>v</sub>}. Besides that, there is definition of reflexive edge strength of <em>G</em>(<em>V</em>,<em>E</em>) denoted as <em>res</em>(<em>G</em>), that is a minimum <em>k</em> that using for labeling <em>f</em> on <em>G</em>(<em>V</em>,<em>E</em>). This paper will discuss about edge irregular reflexive <em>k</em>-labeling for sun graph and corona of cycle and null graph, denoted by <em>C</em><sub>n</sub> ⨀ <em>N</em><sub>2</sub> and make sure about their reflexive edge strengths.</p>
Cited by 5Open access
Athirah Nawawi, Titin Sri Martini, Tri Atmojo Kusmayadı · AIP conference proceedings · 2025
Cited by 0