Regular Graphs Research Articles

Incremental semi-supervised graph learning NMF with block-diagonal

Non-negative matrix factorization (NMF), as a good data dimensionality reduction method, is widely used in the field of image recognition. Incremental non-negative matrix factorization (INMF) as an improvement solves the problem of inefficiency caused by repeated running of data samples during online learning. However, in the traditional incremental non-negative matrix factorization algorithm, the newly added training samples do not contain label information. Some samples may be marked in both the initial sample and the new training sample in the real application scenario. In order to make full use of the label information carried by the dataset, in this paper, we propose a semi-supervised non-negative matrix factorization model for batch incremental data, incremental semi-supervised graph learning NMF with block diagonal (ISGDNMF). The model is divided into three cases according to the degree of label-carrying for the new batch data: all label-carrying, no label-carrying, and partial label-carrying. The label information is also used to add a diagonal structure to the coefficient matrix, which makes it possible to have stronger discriminatory ability and to distinguish different classes of images more easily. And graph regularization information is added in order to maintain the spatial-geometric structure of the data. Experiments on six image datasets show that this algorithm has superior performance relative to the other seven NMF-based algorithms.

Total difference chromatic numbers of regular infinite graphs

Total difference chromatic numbers of regular infinite graphs

Max-min greedy matching problem: Hardness for the adversary and fractional variant

Eden, Feige, and Feldman explored the concept of the max-min greedy matching problem, which can be conceived as a game played between an algorithm and an adversary. Both parties are presented with a bipartite graph consisting of items and players. Initially, the algorithm selects a priority order for the items, while the adversary, in turn, chooses a priority order for the players based on the algorithm's choice. These priority orders are then utilized in a greedy process to establish a matching between the items and players. During the process, when it is a player's turn, they select the highest priority item from their remaining available neighbors. The objective of the algorithm is to maximize the size of the resulting matching, while the adversary's goal is to minimize its size. Previous studies have demonstrated that the algorithm can employ a polynomial-time strategy to achieve a competitive ratio greater than 12.In this research, we present evidence that, from the adversary's standpoint, approximating the adversarial order minimum matching problem with a ratio better than 65 is NP-hard, assuming the small set expansion (SSE) hypothesis. Conversely, we introduce a fractional variant of the problem and analyze the interaction between the algorithm and the adversary when either or both parties are allowed to utilize fractional permutations. Notably, we discover that if the algorithm solely employs integral item permutations, then an optimal response from the adversary can also involve integral player permutations. Additionally, we establish that, in the fractional variant, the algorithm can adopt a round-robin strategy to achieve a competitive ratio of at least 1−1/e for input graphs with a sufficiently large granularity parameter m. Furthermore, we demonstrate that the analysis for the round-robin strategy is tight even for regular graphs.

Majority vote in social networks

Consider a graph G and suppose that initially each node is colored either black or white. In the majority model, in each round all nodes simultaneously update their color to the most frequent color among their neighbors.Experiments on the graph data from the real world social networks (SNs) suggest that if an extremely small set of high-degree nodes, often referred to as the elites, all agree on a color, that color becomes the dominant color at the end of the process. We propose two countermeasures that can be adopted by individual nodes relatively easily and guarantee that the elites will not have this disproportionate power to engineer the dominant output color. The first countermeasure essentially requires each node to make some new connections at random, while the second one demands the nodes to be more reluctant towards changing their color. We verify their effectiveness and correctness both theoretically and experimentally.We also investigate the majority model and a variant of it when the initial coloring is random on the real world SNs and several random graph models. In particular, our results on the Erdős-Rényi and regular random graphs confirm or support several theoretical findings or conjectures by the prior work regarding the threshold behavior of the process.

Heat-Bath and Metropolis Dynamics in Ising-like Models on Directed Regular Random Graphs.

Using a single-site mean-field approximation (MFA) and Monte Carlo simulations, we examine Ising-like models on directed regular random graphs. The models are directed-network implementations of the Ising model, Ising model with absorbing states, and majority voter models. When these nonequilibrium models are driven by the heat-bath dynamics, their stationary characteristics, such as magnetization, are correctly reproduced by MFA as confirmed by Monte Carlo simulations. It turns out that MFA reproduces the same result as the generating functional analysis that is expected to provide the exact description of such models. We argue that on directed regular random graphs, the neighbors of a given vertex are typically uncorrelated, and that is why MFA for models with heat-bath dynamics provides their exact description. For models with Metropolis dynamics, certain additional correlations become relevant, and MFA, which neglects these correlations, is less accurate. Models with heat-bath dynamics undergo continuous phase transition, and at the critical point, the power-law time decay of the order parameter exhibits the behavior of the Ising mean-field universality class. Analogous phase transitions for models with Metropolis dynamics are discontinuous.

Construction of Simultaneous Cospectral Graphs for Adjacency, Laplacian and Normalized Laplacian Matrices

In this paper we construct several classes of non-regular graphs which are co-spectral with respect to all the three matrices, namely, adjacency, Laplacian and normalized Laplacian, and hence we answer a question asked by Butler [?]. We make these constructions starting with two pairs (G1, H1) and (G2, H2) of A-cospectral regular graphs, then considering the subdivision graphs S(Gi) and R-graphs ℛ(Hi), i = 1, 2, and finally making some kind of partial joins between S(G1) and ℛ(G2) and S(H1) and ℛ(H2). Moreover, we determine the number of spanning trees and the Kirchhoff index of the newly constructed graphs.

A New Approach on Bipolar Second Degree Perfect Fuzzy Matching for Bipolar Fuzzy Graphs Based on Vertices

Graph is a visual representation of data. In mathematics, graph is a structure which creates the pairwise relation between objects. The objects are known as nodes or vertices and the connection between two nodes are known as links or edges. A graph is a symmetric binary relation on a non-empty set whereas Fuzzy graph is a symmetric binary fuzzy relation on a fuzzy subset. Bipolar Fuzzy graph is a generalization of graph theory by using bipolar Fuzzy set. The bipolar Fuzzy sets are the extension of Fuzzy sets. Our main objective is to introduce Bipolar Second Degree Perfect Fuzzy Matching for Bipolar Fuzzy Graph based on vertices. Secondly, we proved the necessary conditions under which they are equivalent and also proved that for a particular condition a perfect second-degree fuzzy matching is not a regular bipolar fuzzy graph. Next, we discussed about perfect second-degree Fuzzy matching and second degree fuzzy matching number for bipolar fuzzy graph on the cycle.

Hitting all maximum stable sets in P5-free graphs

We prove that every P5-free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where Pt denotes the t-vertex path, and for graphs G,H, we say G is H-free if no induced subgraph of G is isomorphic to H).More generally, let us say a class C of graphs is η-bounded if there exists a function h:N→N such that η(G)≤h(ω(G)) for every graph G∈C, where η(G) denotes smallest cardinality of a hitting set of all maximum stable sets in G, and ω(G) is the clique number of G. Also, C is said to be polynomially η-bounded if in addition h can be chosen to be a polynomial.We introduce η-boundedness inspired by a question of Alon (asking how large η(G) can be for a 3-colourable graph G), and motivated by a number of meaningful similarities to χ-boundedness, namely,•given a graph G, we have η(H)≤ω(H) for every induced subgraph H of G if and only if G is perfect;•there are graphs G with both η(G) and the girth of G arbitrarily large; and•if C is a hereditary class of graphs which is polynomially η-bounded, then C satisfies the Erdős-Hajnal conjecture. The second bullet above in particular suggests an analogue of the Gyárfás-Sumner conjecture, that the class of all H-free graphs is η-bounded if (and only if) H is a forest. Like χ-boundedness, the case where H is a star is easy to verify, and we prove two non-trivial extensions of this: H-free graphs are η-bounded if (1) H has a vertex incident with all edges of H, or (2) H can be obtained from a star by subdividing at most one edge, exactly once.Unlike χ-boundedness, the case where H is a path is surprisingly hard. Our main result mentioned at the beginning shows that P5-free graphs are η-bounded. The proof is rather involved compared to the classical “Gyárfás path” argument which establishes, for all t, the χ-boundedness of Pt-free graphs. It remains open whether Pt-free graphs are η-bounded for t≥6.It also remains open whether P5-free graphs are polynomially η-bounded, which, if true, would imply the Erdős-Hajnal conjecture for P5-free graphs. But we prove that H-free graphs are polynomially η-bounded if H is a proper induced subgraph of P5. We further generalize the case where H is a 1-regular graph on four vertices, showing that H-free graphs are polynomially η-bounded if H is a forest with no vertex of degree more than one and at most four vertices of degree one.

On random irregular subgraphs

AbstractLet be a ‐regular graph on vertices. Frieze, Gould, Karoński, and Pfender began the study of the following random spanning subgraph model . Assign independently to each vertex of a uniform random number , and an edge of is an edge of if and only if . Addressing a problem of Alon and Wei, we prove that if , then with high probability, for each nonnegative integer , there are vertices of degree in .

AN INTRO TO $\delta_d$-FUZZY GRAPHS

Graph is a easy way to represent the real life situation. Graph is a combination of Points and Lines. In network analysis, the degree of a point plays a prominent role in Graph Theory. The degree of a point is the number of connections it has with the other points in the point set. Among the degrees of all the points in graph $G^*$, the minimum value is denoted by $\delta(G^*)$. In this article, a new abstraction of fuzzy graph is initiated by combining the parameters, degree of a point and minimum degree of the graph and termed it is as $\delta_d$-fuzzy graphs. Order and Size on $\delta_d$-fuzzy graphs were studied and Handshaking Lemma were explained with illustration. Idea on $\delta_d$-regular fuzzy graph were interpreted using the theorems. Also operations on graphs such as union, intersection, complement, cartesian product, Tensor Product, Corona are extended for $\delta_d$-fuzzy graphs.

On the complexity of co-secure dominating set problem

A set D⊆V of a graph G=(V,E) is a dominating set of G if every vertex v∈V∖D is adjacent to at least one vertex in D. A set S⊆V is a co-secure dominating set (CSDS) of a graph G if S is a dominating set of G and for each vertex u∈S there exists a vertex v∈V∖S such that uv∈E and (S∖{u})∪{v} is a dominating set of G. The minimum cardinality of a co-secure dominating set of G is the co-secure domination number and it is denoted by γcs(G). Given a graph G=(V,E), the minimum co-secure dominating set problem (Min Co-secure Dom) is to find a co-secure dominating set of minimum cardinality. In this paper, we strengthen the inapproximability result of Min Co-secure Dom for general graphs by showing that this problem can not be approximated within a factor of (1−ε)ln⁡|V| for perfect elimination bipartite graphs and star convex bipartite graphs unless P=NP. On the positive side, we show that Min Co-secure Dom can be approximated within a factor of O(ln⁡|V|) for any graph G with δ(G)≥2. For 3-regular and 4-regular graphs, we show that Min Co-secure Dom is approximable within a factor of 83 and 103, respectively. Furthermore, we prove that Min Co-secure Dom is APX-complete for 3-regular graphs.

Some results on the Wiener index related to the Šoltés problem of graphs

The Wiener index, W(G), of a connected graph G is the sum of distances between its vertices. In 2021, Akhmejanova et al. posed the problem of finding graphs G with large Rm(G)=|{v∈V(G)∣W(G)−W(G−v)=m∈Z}|/|V(G)|. It is shown that there is a graph G with Rm(G)>1/2 for any integer m≥0. In particular, there is a regular graph of even degree with this property for any odd m≥1. The proposed approach allows to construct new families of graphs G with R0(G)→1/2 when the order of G increases.

Centric graph regularized log-norm sparse non-negative matrix factorization for multi-view clustering

Multi-view non-negative matrix factorization (NMF) provides a reliable method to analyze multiple views of data for low-dimensional representation. A variety of multi-view learning methods have been developed in recent years, demonstrating successful applications in clustering. However, existing methods in multi-view learning often tend to overlook the non-linear relationships among data and the significance of the similarity of internal views, both of which are essential in multi-view tasks. Meanwhile, the mapping between the obtained representation and the original data typically contains complex hidden information that deserves to be thoroughly explored. In this paper, a novel multi-view NMF is proposed that explores the local geometric structure among multi-dimensional data and learns the hidden representation of different attributes through centric graph regularization and pairwise co-regularization of the coefficient matrix. In addition, the proposed model is further sparsified with l2,log-(pseudo) norm to efficiently generate sparse solutions. As a result, the model obtains a better part-based representation, enhancing its robustness and applicability in complex noisy scenarios. An effective iterative update algorithm is designed to solve the proposed model, and the convergence of the algorithm is proven to be theoretically guaranteed. The effectiveness of the proposed method is verified by comparing it with nine state-of-the-art methods in clustering tasks of eight public datasets.

Simple versus nonsimple loops on random regular graphs

AbstractIn this note, we solve the “birthday problem” for loops on random regular graphs. Namely, for fixed , we prove that on a random ‐regular graph with vertices, as approaches infinity, with high probability: (i) almost all primitive nonbacktracking loops of length are simple, that is, do not self‐intersect, and (ii) almost all primitive nonbacktracking loops of length self‐intersect.

Joint orthogonal symmetric non-negative matrix factorization for community detection in attribute network

Community detection is an important and challenging task in complex attribute network analysis. Symmetric non-negative matrix factorization-based methods have become promising because of their excellent ability to extract low-dimensional representations of attribute networks (which are characterized by their adjacency and attribute data). In this paper, we propose a novel community detector called joint symmetric non-negative matrix factorization model, in which the attribute homogeneity and topology similarities of an attribute network are characterized in a unified framework. An orthogonal constraint is imposed on the factor matrix to improve the accuracy of nodes’ affiliations to communities. Furthermore, a novel multi-order graph regularization was developed to preserve the intrinsic geometric structure, and an eigenvector centrality-based enhancement strategy was established to explore more comprehensive adjacency information. Extensive experiments on community detection tasks demonstrate that the proposed method performs significantly better than existing state-of-the-art methods in most benchmark complex networks.

Full-homomorphisms to paths and cycles

A full-homomorphism between a pair of graphs is a vertex mapping that preserves adjacencies and non-adjacencies. For a fixed graph H, a full H-colouring is a full-homomorphism of G to H. A minimal H-obstruction is a graph that does not admit a full H-colouring, such that every proper induced subgraph of G admits a full H-colouring. Feder and Hell proved that for every graph H there is a finite number of minimal H-obstructions. We begin this work by describing all minimal obstructions of paths. Then, we study minimal obstructions of regular graphs to propose a description of minimal obstructions of cycles. As a consequence of these results, we observe that for each path P and each cycle C, the number of minimal P-obstructions and C-obstructions is O(|V(P)|2) and O(|V(C)|2), respectively. Finally, we propose some problems regarding the largest minimal H-obstructions, and the number of minimal H-obstructions.

Short Proof of the Asymptotic Confirmation of the Faudree-Lehel Conjecture

Given a simple graph $G$, the irregularity strength of $G$, denoted $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ for which each vertex weight $f^V(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$ is unique amongst all $v\in V(G)$. In 1987, Faudree and Lehel conjectured that there is a constant $c$ such that $s(G) \leq n/d + c$ for all $d$-regular graphs $G$ on $n$ vertices with $d>1$, whereas it is trivial that $s(G) \geq n/d$. In this short note we prove that the Faudree-Lehel Conjecture holds when $d \geq n^{0.8+\epsilon}$ for any fixed $\epsilon >0$, with a small additive constant $c=28$ for $n$ large enough. Furthermore, we confirm the conjecture asymptotically by proving that for any fixed $\beta\in(0,1/4)$ there is a constant $C$ such that for all $d$-regular graphs $G$, $s(G) \leq \frac{n}{d}(1+\frac{C}{d^{\beta}})+28$, extending and improving a recent result of Przybyło that $s(G) \leq \frac{n}{d}(1+ \frac{1}{\ln^{\epsilon/19}n})$ whenever $d\in [\ln^{1+\epsilon} n, n/\ln^{\epsilon}n]$ and $n$ is large enough.

The Structural Properties of (2, 6)-Fullerenes

A (2,6)-fullerene F is a 2-connected cubic planar graph whose faces are only 2-length and 6-length. Furthermore, it consists of exactly three 2-length faces by Euler’s formula. The (2,6)-fullerene comes from Došlić’s (k,6)-fullerene, a 2-connected 3-regular plane graph with only k-length faces and hexagons. Došlić showed that the (k,6)-fullerenes only exist for k=2, 3, 4, or 5, and some of the structural properties of (k,6)-fullerene for k=3, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of (2,6)-fullerene. We discover that the edge-connectivity of (2,6)-fullerenes is 2. Every (2,6)-fullerene is 1-extendable, but not 2-extendable (F is called n-extendable (|V(F)|≥2n+2) if any matching of n edges is contained in a perfect matching of F). F is said to be k-resonant (k≥1) if the deleting of any i (0≤i≤k) disjoint even faces of F results in a graph with at least one perfect matching. We have that every (2,6)-fullerene is 1-resonant. An edge set, S, of F is called an anti−Kekulé set if F−S is connected and has no perfect matchings, where F−S denotes the subgraph obtained by deleting all edges in S from F. The anti−Kekulé number of F, denoted by ak(F), is the cardinality of a smallest anti−Kekulé set of F. We have that every (2,6)-fullerene F with |V(F)|>6 has anti−Kekulé number 4. Further we mainly prove that there exists a (2,6)-fullerene F having fF hexagonal faces, where fF is related to the two parameters n and m.

WSNMF: Weighted Symmetric Nonnegative Matrix Factorization for attributed graph clustering

In recent times, Symmetric Nonnegative Matrix Factorization (SNMF), a derivative of Nonnegative Matrix Factorization (NMF), has surfaced as a promising technique for graph clustering. Nevertheless, when applied to attributed graph clustering, it confronts notable challenges. These include the disregard for attributed information, the oversight of geometric data point structures, and the inability to discriminate irrelevant features and data outliers. In response, we introduce an innovative extension of SNMF termed Weighted Symmetric Nonnegative Matrix Factorization (WSNMF). This method introduces node attribute similarity to compute a weight matrix, effectively bridging the gap for attributed graph clustering. Our approach incorporates graph regularization and sparsity constraints to uphold the geometric structure of data points and discern irrelevant features and data outliers. Additionally, we present an updating rule to address optimization complexities and validate algorithmic convergence. Rigorous experimentation on real-world and synthetic networks, employing well-established metrics including F-measure, RI, Modularity, Density, and entropy, substantiates the performance enhancement offered by WSNMF.

Some Results on Pendant Regular Graphs

Let G = (V, E) be a simple connected graph with o(G) = n and s(G) = m. A graph with pendant vertices is called Pendant Graphs or simply P-Graphs. In this paper we define the regularity of Pendant Graphs w.r.t. the degrees of support vertex of each pendant vertices. Sufficient conditions for pendant graphs are discussed in this paper.