Class Of Graphs Research Articles

Local certification of graph decompositions and applications to minor-free classes

Local certification consists in assigning labels to the vertices of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received considerable attention. The goal is to certify that a graph G belongs to a given graph class G. Such certifications with labels of size O(log⁡n) (where n is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors.In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log⁡n). We also show matching lower bounds using a new proof technique.

The continuous-time magnetic quantum walk and its probability invariance on a class of graphs

Let [Formula: see text] be a nonnegative integer and [Formula: see text] the power set of the set [Formula: see text]. Then there is an adjacency relation in [Formula: see text] such that [Formula: see text] together with the relation forms a regular graph. In this paper, we propose a model of continuous-time magnetic quantum walk (MQW) on the graph [Formula: see text], and investigate its properties from a viewpoint of probability and quantum information. We first introduce a magnetic Laplacian [Formula: see text] on the graph [Formula: see text] and examine its spectrum. And then, with [Formula: see text] as the Hamiltonian, we construct our model of continuous-time MQW on the graph [Formula: see text]. We find that the model has probability distributions that are completely independent of the magnetic potential at all times. And we show that it has perfect state transfer at time [Formula: see text] when the magnetic potential satisfies some mild conditions. Some other interesting results are also obtained.

Approximation algorithms for maximum weighted internal spanning trees in regular graphs and subdivisions of graphs

Abstract Let $G$ be a vertex-weighted connected graph of $n$ vertices and let $T$ be a spanning tree of $G$. We call $T$ a maximum weighted internal spanning tree of $G$ if the sum of the weights of the internal vertices of $T$ is the maximum over all spanning trees of $G$. The maximum weighted internal spanning tree (MaxwIST) problem asks to find such a spanning tree $T$ of $G$. The problem is NP-hard. We give an $O(dn)$ time approximation algorithm for $d$-regular graphs of $n=|V|$ vertices that computes a spanning tree with total weight of the internal vertices is at least $\frac{\beta _{d}}{\beta _{d} +d-2} - \epsilon $ of the total weight of all the vertices of the graph for any $\epsilon>0$, where $\beta _{d} = (d-1)H_{d-1}$, and $H_{d-1} = \sum _{i=1}^{d-1} i^{-1}$ is the $(d-1)$th harmonic number. For every $d \geq 3$ and $n_{0} \geq 1$, we show the construction of a $d$-regular graph of at least $n_{0}$ vertices, such that for any of its spanning trees, $\frac{w(I)}{w(V)}\le \frac{d}{d+1}$ holds. We give an $O(dn)$ time approximation algorithm for subdivisions of $d$-regular graphs, where the ratio of the internal weight of the spanning tree with the total vertex weight of the graph is at least $\frac{d-1}{2d-3} - \epsilon $ for $\epsilon>0$. We extend our study to $x$-subdivisions of Hamiltonian and hypoHamiltonian graphs, where each edge of the original Hamiltonian or hypoHamiltonian graph has been subdivided at least $x$ times. For those two graph classes, we show that there exists a spanning tree with internal vertex weight at least $1-\frac{2}{x-1}$ of the total vertex weight of the graph. Furthermore, we give $O(n)$ time algorithm for $x$-subdivisions of biconnected outerplanar graphs and $4$-connected planar graphs to achieve the above bound.

MDGCL: Graph Contrastive Learning Framework with Multiple Graph Diffusion Methods

In recent years, some classical graph contrastive learning(GCL) frameworks have been proposed to address the problem of sparse labeling of graph data in the real world. However, in node classification tasks, there are two obvious problems with existing GCL frameworks: first, the stochastic augmentation methods they adopt lose a lot of semantic information; second, the local–local contrasting mode selected by most frameworks ignores the global semantic information of the original graph, which limits the node classification performance of these frameworks. To address the above problems, this paper proposes a novel graph contrastive learning framework, MDGCL, which introduces two graph diffusion methods, Markov and PPR, and a deterministic–stochastic data augmentation strategy while retaining the local–local contrasting mode. Specifically, before using the two stochastic augmentation methods (FeatureDrop and EdgeDrop), MDGCL first uses two deterministic augmentation methods (Markov diffusion and PPR diffusion) to perform data augmentation on the original graph to increase the semantic information, this step ensures subsequent stochastic augmentation methods do not lose too much semantic information. Meanwhile, the diffusion matrices carried by the augmented views contain global semantic information of the original graph, allowing the framework to utilize the global semantic information while retaining the local-local contrasting mode, which further enhances the node classification performance of the framework. We conduct extensive comparative experiments on multiple benchmark datasets, and the results show that MDGCL outperforms the representative baseline frameworks on node classification tasks. Among them, compared with COSTA, MDGCL’s node classification accuracy has been improved by 1.07% and 0.41% respectively on two representative datasets, Amazon-Photo and Coauthor-CS. In addition, we also conduct ablation experiments on two datasets, Cora and CiteSeer, to verify the effectiveness of each improvement work of our framework.

Extremal graphs with respect to modified Sombor index

The modified Sombor (mSO) index of a given graph [Formula: see text] is symbolized with mSO and define by sum of the weights [Formula: see text], where [Formula: see text] is the degree of a vertex [Formula: see text] in [Formula: see text]. In this work, we present the first and second maximum values of the mSO index among all [Formula: see text]-vertex [Formula: see text]-cyclic graphs. We also provide that the mSO index attains its maximum and second maximum on two classes of chemical graphs. Finally, we will present new lower and upper bounds for the mSO index of connected chemical graphs.

Theoretical studies of the [formula omitted]-strong Roman domination problem

The concept of Roman domination has been a subject of intrigue for more than two decades, with the fundamental Roman domination problem standing out as one of the most significant challenges in this field. This article studies a practically motivated generalization of this problem, known as the k-strong Roman domination. In this problem variant, defenders within a network are tasked with safeguarding any k vertices simultaneously, under multiple attacks. The aim is to find a feasible mapping that assigns a (integer) weight to each vertex of the input graph with a minimum sum of weights across all vertices. A function is considered feasible if any non-defended vertex, i.e. one labeled by zero, is protected by at least one of its neighboring vertices labeled by at least two. Furthermore, each defender ensures the safety of a non-defended vertex by imparting a value of one to it while retaining a one for themselves. To the best of our knowledge, this paper represents the first theoretical study on this problem. The study presents results for general graphs, establishes connections between the problem at hand and other domination problems, and provides exact values and bounds for specific graph classes, including complete graphs, paths, cycles, complete bipartite graphs, grids, and a few selected classes of convex polytopes. Additionally, an attainable lower bound for general cubic graphs is provided.

Metric dimension of unit graphs

The notion of the metric dimension of a graph is well-known and its study is well entrenched in the literature. In this paper, we introduce new classes of graphs that exhibit remarkable characteristic: the metric dimension and the diameter of the unit graph are equal. Additionally, we provide a characterization of finite commutative rings [Formula: see text], wherein the metric dimension assumes a value [Formula: see text], where [Formula: see text]. Further, we also demonstrate an exhaustive examination that ascertains the precise finite commutative rings [Formula: see text], in which domination number is equal to metric dimension of unit graph.

Distributed Independent Sets in Interval and Segment Intersection Graphs

The Maximum Independent Set problem is well-studied in graph theory and related areas. An independent set of a graph is a subset of non-adjacent vertices of the graph. A maximum independent set is an independent set of maximum size. This paper studies the Maximum Independent Set problem in some classes of geometric intersection graphs in a distributed setting. More precisely, we study the Maximum Independent Set problem on two geometric intersection graphs, interval and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in a model that is similar but a little weaker than the local communication model. We compute the maximum independent set on interval graphs in [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] is the size of the maximum independent set and [Formula: see text] is the number of nodes in the graph. We provide a matching lower bound of [Formula: see text] on the number of rounds, whereas [Formula: see text] is a trivial lower bound on message complexity. Thus, our algorithm is both time and message-optimal. We also study the Maximum Independent Set problem in interval count [Formula: see text] graphs, a special case of the interval graphs where the intervals have exactly [Formula: see text] different lengths. We propose an [Formula: see text]-approximation algorithm that runs in [Formula: see text] round. For axis-parallel segment intersection graphs, we design an [Formula: see text]-approximation algorithm that obtains a solution in [Formula: see text] rounds. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].

Leanness Computation: Small Values and Special Graph Classes

Let u and v be vertices in a connected graph G = (V, E). For any integer k such that 0 ≤ k ≤ dG (u, v), the k-slice Sk (u, v) contains all vertices x on a shortest uv-path such that dG (u, x) = k. The leanness of G is the maximum diameter of a slice. This metric graph invariant has been studied under different names, such as "interval thinness" and "fellow traveler property". Graphs with leanness equal to 0, a.k.a. geodetic graphs, also have received special attention in Graph Theory. The practical computation of leanness in real-life complex networks has been studied recently (Mohammed et al., COMPLEX NETWORKS'21). In this paper, we give a finer-grained complexity analysis of two related problems, namely: deciding whether the leanness of a graph G is at most some small value ℓ; and computing the leanness on specific graph classes. We obtain improved algorithms in some cases, and time complexity lower bounds under plausible hypotheses.

A semantic backdoor attack against graph convolutional networks

Graph convolutional networks (GCNs) have been very effective in addressing the issue of various graph-structured related tasks, such as node classification and graph classification. However, recent research has shown that GCNs are vulnerable to a new type of threat called a backdoor attack, where the adversary can inject a hidden backdoor into GCNs so that the attacked model performs well on benign samples, but its prediction will be maliciously changed to the attacker-specified target class if the hidden backdoor is activated by the attacker-defined trigger. A semantic backdoor attack is a new type of backdoor attack on deep neural networks (DNNs), where a naturally occurring semantic feature of samples can serve as a backdoor trigger such that the infected DNNs models will misclassify testing samples containing the predefined semantic feature even without the requirement of modifying the testing samples. Since the backdoor trigger is a naturally occurring semantic feature of the samples, semantic backdoor attacks are more imperceptible and pose a new and serious threat. Existing research on semantic backdoor attacks focuses on the tasks of CNNs-based (Convolutional Neural Networks) image classification and LSTM-based (Long Short-Term Memory) text classification or word prediction. Little attention has been given to semantic backdoor attacks on GCNs models.In this paper, we investigate whether such semantic backdoor attacks are possible for GCNs and propose a semantic backdoor attack against GCNs (SBAG) under the context of graph classification to reveal the existence of this security vulnerability in GCNs. SBAG uses a certain type of nodes in the samples as a backdoor trigger and injects a hidden backdoor into GCNs models by poisoning training data. The backdoor will be activated, and the GCNs models will give malicious classification results specified by the attacker even on unmodified samples as long as the samples contain enough trigger nodes. We evaluate SBAG on five graph datasets. The experimental results indicate that SBAG can achieve attack success rates of approximately 99.9% on unmodified testing samples that naturally contain the trigger and attack success rates over 82% on testing samples modified to inject the trigger, respectively, both under poisoning rates of less than 5%. 11Our code is available at: github.

IC points weight learning-based GCN and improving feature distribution for industrial fault diagnosis

Industrial fault diagnosis (FD) is crucial to detect the causes of faults in a timely manner and solve the problems to maintain stable production. Adequately considering the spatial structure of the samples and their features is an urgent problem for better FD. This paper focused on the points located in the within-class boundaries and between-class intersection region (IC points) and proposed IC point weight learning-based graph convolution network (GCN) and improving feature distribution (ICWGCN-FD) method. Firstly, ICWGCN-FD assigns appropriate weights to IC points specially allocated by graph learning layer. Secondly, a cosine distance loss term is particularly designed to make the IC points closer to the center of their respective classes and further away from their nearest neighbor points belongs to other classes. Finally, the spatial feature distribution of the overall samples is improved under the influence of the message passing function possessed by GCN, thus presenting a clearer distribution. Two cases from industrial simulation are carried out to validate the practical feasibility and superiority of ICWGCN-FD, compared with other classic and recent graph neural networks. In addition, the 3D visualization results correspond to the principles of the method, and an ablation experiment successfully verified the validity of each component.

A Liouville theorem for elliptic equations with a potential on infinite graphs

We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is u≡0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u\\equiv 0$$\\end{document}. We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions.

LightCapsGNN: light capsule graph neural network for graph classification

LightCapsGNN: light capsule graph neural network for graph classification

A characterization for fuzzy strong cut vertices and fuzzy strong cut edges

Cut vertices and cut edges are valuable for analyzing connectivity problems in classical graph theory. However, they cannot deal with certain fuzzy problems. In order to solve this problem, this paper introduces the definitions of fuzzy strong cut vertices and fuzzy strong cut edges, and characterizes fuzzy strong cut vertices and fuzzy strong cut edges in fuzzy trees, complete fuzzy graphs, and fuzzy cycles. Finally, practical applications verify the effectiveness of the theory in network stability analysis.

Reverse Zagreb Indices and Their Application in the Evaluation of Physiochemical Properties of Anticancer/Antibacterial Drugs.

Recent advancements in chemical graph theory have facilitated a deeper understanding of the relationship between chemical structures and their underlying graphs based upon classical graph theory principles. Quantitative structure-property relationship (QSPR) analysis stands out as a powerful tool for probing chemical graphs. Topological indices, graph invariants assigning numerical values to graphs, play a pivotal role in statistically correlating physical properties, chemical reactivity, and biological activity across diverse chemical structures. The cactus graph (CG) represents a noteworthy family of connected graphs distinguished by the absence of common vertices between cycles. In this study, we focus on establishing the expressions of Zagreb and reverse Zagreb indices in terms of known parameters for cactus graphs. Subsequently, we leverage these indices to conduct QSPR analysis, employing linear and multilinear regression models to investigate the relationship between the properties of chemical structures adorned with cactus-like graphs and the corresponding Zagreb and reverse Zagreb indices.

On Zagreb Energy of Certain Classes of Graphs

On Zagreb Energy of Certain Classes of Graphs

Orientations without forbidden patterns on three vertices

Given a set F of oriented graphs, a graph G is a Forbe(F)-graph if it admits an F-free orientation. Skrien showed that proper-circular arc graphs, nested interval graphs and comparability graphs, correspond to Forbe(F)-graph classes for some set F of orientations of P3. Building on these results, we exhibit the list of all Forbe(F)-graph classes when F is a set of oriented graphs on three vertices. Structural characterizations for these classes are provided, except for the so-called perfectly-orientable graphs and the transitive-perfectly-orientable graphs, which remain as open problems.

On two variants of split graphs: 2-unipolar graph and k-probe-split graph

A graph is called split if its vertex set can be partitioned into a stable set and a clique. In this article, we studied two variants of split graphs. A graph G is polar if its vertex set can be partitioned into two sets A and B such that G[A] is a complete multipartite graph and G[B] is a disjoint union of complete graphs. A 2-unipolar graph is a polar graph G such that G[A] is a clique and G[B] is the disjoint union of complete graphs with at most two vertices. We present a minimal forbidden induced subgraph characterization for 2-unipolar graphs. In addition, we show that they can be represented as an intersection of substars of special cacti. Let G be a graph class, the G-width of a graph G is the minimum positive integer k such that there exist k independent sets N1, … , Nk such that a set F of nonedges of G, whose endpoints belong to some Ni with i = 1, … , k, can be added so that the resulting graph G′ belongs to G. We say that a graph G is k-probe-G if it has G-width at most k and when G is the class of split graphs it is denominated k-probe-split. We prove that deciding, given a graph G and a positive integer k, whether G is a h-probe-split graph for some h ≤ k is NP-complete. Besides, a characterization by minimal forbidden induced subgraphs for 2-probe-split cographs is presented.

Some spectral comparison results on infinite quantum graphs

In this paper we establish spectral comparison results for Schrödinger operators on a certain class of infinite quantum graphs, using recent results obtained in the finite setting. We also show that new features do appear on infinite quantum graphs such as a modified local Weyl law. In this sense, we regard this paper as a starting point for a more thorough investigation of spectral comparison results on more general infinite metric graphs.

APOE, Immune Factors, Sex, and Diet Interact to Shape Brain Networks in Mouse Models of Aging.

Current interventions for Alzheimer's disease (AD) do not provide a cure, and are delivered years after neuropathological onset. Addressing the impact of risk factors on brain networks holds promises for early detection, prevention, and revealing putative therapeutic targets at preclinical stages. We utilized six mouse models to investigate the impact of factors, including APOE genotype, age, sex, immunity, and diet, on brain networks. Large structural connectomes were derived from high resolution compressed sensing diffusion MRI. A highly parallelized graph classification identified subnetworks associated with unique risk factors, revealing their network fingerprints, and a common network composed of 63 connections with shared vulnerability to all risk factors. APOE genotype specific immune signatures support the design of interventions tailored to risk profiles.