Vertex Connectivity Research Articles

On maximum degree (signless) Laplacian matrix of a graph

Let G be a simple graph on n vertices and v1, v2,...,vn be the vértices of G. We denote the degree of a vertex vi in G by dG(vi) = di. The maximum degree matrix of G, denoted by M(G), is the real symmetric matrix with its ijth entry equal to max{di, dj} if the vertices vi and vj are adjacent in G, 0 otherwise. In analogous to the definitions of Laplacian matrix and signless Laplacian matrix of a graph, we consider Laplacian and signless Laplacian for the maximum degree matrix, called the maximum degree Laplacian matrix and the maximum degree signless Laplacian matrix, respectively. Also, we introduce maximum degree Laplacian energy and maximum degree signless Laplacian energy of a graph. Then we determine the maximum degree (signless) Laplacian energy of some graphs in terms of ordinary energy, and (signless) Laplacian energy. We compute the máximum degree (signless) Laplacian spectra of some graph compositions. A lower and upper bound for the largest eigenvalue of the maximum degree (signless) Laplacian matrix is established and also we determine an upper bound for the second smallest eigenvalue of maximum degree Laplacian matrix in terms of vertex connectivity. We also determine bounds for the maximum degree (signless) Laplacian energy in terms of first Zagreb index.

Rainbow Vertex Antimagic Coloring 2-Connection paada Keluarga Graf Tangga

All graph in this paper are connected graph and simple graph. Let G = (V,E)be a connected graph. Rainbow vertex connection is the assignment of G that has interior vertices with different colors. The minimum number of colors from the rainbow vertex coloring in graph G is called rainbow vertex connection number. If wf(u) ̸= wf(v) for two different vertext u, v ∈ V (G) then f is called antimagic labeling for graph G. Rainbow vertex antimagic coloring is a combination between rainbow coloring and antimagic labeling. Graph G is called rainbow vertex antimagic coloring 2-connection if G has at least 2 rainbow paths from u − v. Rainbow vertex antimagic coloring 2-connection to denoted as rvac2(G). In this paper, we will study rainbow vertex antimagic coloring 2-connection on a family of graphs ladder that includes H-graph Hn for n ≥ 2, slide ladder graph SLn for n ≥ 2, and graph Octa-Chain OCn for n ≥ 2.

Spectrum of Johnson graphs

In this paper we prove new bounds on the second eigenvalue of Johnson graphs. We then apply these bounds to obtain new results on the modularity of Johnson graphs and their random subgraphs, hamiltonicity of Johnson graphs and thresholds of the appearance of the Hamilton cycles and giant components. Furthermore, we provide general bounds on vertex connectivity and the stability of the modularity for (n,d,λ)-graphs.

Connectivity of Triangulation Flip Graphs in the Plane

Given a finite point set P in general position in the plane, a full triangulation of P is a maximal straight-line embedded plane graph on P. A partial triangulation of P is a full triangulation of some subset P' of P containing all extreme points in P. A bistellar flip on a partial triangulation either flips an edge (called edge flip), removes a non-extreme point of degree 3, or adds a point in P setminus P' as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The edge flip graph is defined with full triangulations as vertices, and edge flips determining the adjacencies. Lawson showed in the early seventies that these graphs are connected. The goal of this paper is to investigate the structure of these graphs, with emphasis on their vertex connectivity. For sets P of n points in the plane in general position, we show that the edge flip graph is lceil {n}/{2}-2rceil -vertex connected, and the bistellar flip graph is (n-3)-vertex connected; both results are tight. The latter bound matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points to 3-space and projecting back the lower convex hull), where (n-3)-vertex connectivity has been known since the late eighties through the secondary polytope due to Gelfand, Kapranov, & Zelevinsky and Balinski’s Theorem. For the edge flip-graph, we additionally show that the vertex connectivity is at least as large as (and hence equal to) the minimum degree (i.e., the minimum number of flippable edges in any full triangulation), provided that n is large enough. Our methods also yield several other results: (i) The edge flip graph can be covered by graphs of polytopes of dimension lceil {n}/{2} -2rceil (products of associahedra) and the bistellar flip graph can be covered by graphs of polytopes of dimension n-3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance n-3 in the Hasse diagram of the partial order of partial subdivisions from the trivial subdivision. (iii) All partial triangulations of a point set are regular iff the partial order of partial subdivisions has height n-3. (iv) There are arbitrarily large sets P with non-regular partial triangulations and such that every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular triangulations.

Self-Training Enhanced: Network Embedding and Overlapping Community Detection With Adversarial Learning.

Network embedding (NE) aims to encode the relations of vertices into a low-dimensional space. After NE, we can obtain the learned vectors of vertices that preserve the proximity of network structures for subsequent applications, e.g., vertex classification and link prediction. In existing NE models, they usually exploit the skip-gram with a negative sampling method to optimize their objective functions. Generally, this method learns the vertex representation only from the local connectivity of vertices (i.e., neighbors). However, there is a larger scope of vertex connectivity in real-world scenarios: a vertex may have multifaceted aspects and should belong to overlapping communities. Taking a social network as the overlapping example, a user may subscribe to the channels of politics, economy, and sports simultaneously, but the politics share more common attributes with the economy and less with the sports. In this article, we propose an adversarial learning approach (ACNE) for modeling overlapping communities of vertices. Specifically, we map the association between communities and vertices into an embedding space. Moreover, we take further research on enhancing our ACNE with the following two operations. First, in the initialization stage, we adopt a walking strategy with perception to obtain paths containing more possible boundary vertices to improve overlapping community detection. Then, after representation learning with ACNE, we use soft community assignments from a simple classifier as supervision to update the weights of ACNE. This self-training mechanism referred to as ACNE-ST can help ACNE to achieve better performance. Experimental results demonstrate that the proposed methods, including ACNE and ACNE-ST, can outperform the state-of-the-art models on the subsequent tasks of vertex classification and overlapping community detection.

Bounds on the connectivity of iterated line graphs

For simple connected graphs that are neither paths nor cycles, we define l ( G )=max{ m : G has a divalent path of length m that is not both of length 2 and in a K 3 } , where a divalent path is a path whose internal vertices have degree two in G . Let G be a graph and L n ( G ) be its n -th iterated line graph of G . We use κ e ′( G ) and κ ( G ) for the essential edge connectivity and vertex connectivity of G , respectively. Let G be a simple connected graph that is not a path, a cycle or K 1, 3 , with l ( G )= l ≥ 1 . We prove that (i) for integers s ≥ 1 , κ ′ e ( L l + s ( G )) ≥ 2 s + 2 ; (ii) for integers s ≥ 2 , κ ( L l + s ( G )) ≥ 2 s − 1 + 2 . The bounds are best possible.

Hamiltonian in 5-Connected Graph

One of the branches of mathematics that studies the properties of graphs is graph theory. The purpose of this study is to find out how to prove the Hamiltonian on a 5-connected graph. Through the stages, including modeling a complete graph, as well as modeling a 5-connected graph using vertices and cut edges, it was found that is a graph that satisfies the characteristics of a 5-connected graph. Analysis of Hamiltonian on 5-connected graph that 5-connected graph is Hamilton's invention, but it is not uniquely Hamiltonian because it has more than one Hamiltonian circuit.Keywords: k-connected graph, Hamiltonian, vertex cuts

Protecting Hybrid Information Transmission Network from Natural and Anthropogenic Hazards

A hybrid network, which consists of the sections of communication lines with the transmission of signals of different physical nature on different transmission media, has been considered. Communication lines respond differently to threats, which allows to choose the line with the best performance for the transmission of information. The causal diagram of events that determine the state of the information transmission network, such as changes in emergency/accident-free time intervals, has been presented. The application scheme of the protection measures against dangerous events has been shown. To verify the measures, a matrix of their compliance with typical natural disasters has been developed and relevant examples have been given. It is suggested to evaluate the flexibility of the telecommunication network by its connectivity, characterized by the numbers of vertex and edge connectivity, the probability of connectivity. The presented scheme of the device for multi-channel information transmission in a hybrid network allows the choice for the transmission of information to the channel with the best performance. Using this device is the essence of the suggestion about increasing the flexibility of the network.

Partitioning a graph into a cycle and a sparse graph

In this paper we investigate results of the form “every graph G has a cycle C such that the induced subgraph of G on V(G)∖V(C) has small maximum degree.” Such results haven't been studied before, but are motivated by the Bessy and Thomassé Theorem which states that the vertices of any graph G can be covered by a cycle C1 in G and disjoint cycle C2 in the complement of G.There are two main theorems in this paper. The first is that every graph has a cycle with Δ(G[V(G)∖V(C)])≤12(|V(G)∖V(C)|−1). The bound on the maximum degree Δ(G[V(G)∖V(C)]) is best possible. The second theorem is that every k-connected graph G has a cycle with Δ(G[V(G)∖V(C)])≤1k+1|V(G)∖V(C)|+3. We also give an application of this second theorem to a conjecture about partitioning edge-coloured complete graphs into monochromatic cycles.

Pyramid Spatial-Temporal Graph Transformer for Skeleton-Based Action Recognition

Although graph convolutional networks (GCNs) have shown their demonstrated ability in skeleton-based action recognition, both the spatial and the temporal connections rely too much on the predefined skeleton graph, which imposes a fixed prior knowledge for the aggregation of high-level semantic information via the graph-based convolution. Some previous GCN-based works introduced dynamic topology (vertex connection relationships) to capture flexible spatial correlations from different actions. Then, the local relationships from both the spatial and temporal domains can be captured by diverse GCNs. This paper introduces a more straightforward and more effective backbone to obtain the spatial-temporal correlation between skeleton joints with a local-global alternation pyramid architecture for skeleton-based action recognition, namely the pyramid spatial-temporal graph transformer (PGT). The PGT consists of four stages with similar architecture but different scales: graph embedding and transformer blocks. We introduce two kinds of transformer blocks in our work: the spatial-temporal transformer block and joint transformer block. In the former, spatial-temporal separated attention (STSA) is proposed to calculate the connection of the global nodes of the graph. Due to the spatial-temporal transformer block, self-attention can be performed on skeleton graphs with long-range temporal and large-scale spatial aggregation. The joint transformer block flattens the tokens in both the spatial and temporal domains to jointly capture the overall spatial-temporal correlations. The PGT is evaluated on three public skeleton datasets: the NTU RGBD 60, NTU RGBD 120 and NW-UCLA datasets. Better or comparable performance with the state of the art (SOTA) shows the effectiveness of our work.

Cyclic Vertex (Edge) Connectivity of Burnt Pancake Graphs

The cyclic vertex (resp., edge) connectivity of a graph [Formula: see text], denoted by [Formula: see text] (resp., [Formula: see text]), is the minimum number of vertices (resp., edges) whose removal from [Formula: see text] results in a disconnected graph and at least two remaining components contain cycles. Thus, to determine the exact values of [Formula: see text] and [Formula: see text] is important in the reliability assessment of interconnection networks. However, the study of the cyclic vertex (edge) connectivity is less involved. In this paper, we determine the cyclic vertex (edge) connectivity of the burnt pancake graphs [Formula: see text] which is the Cayley graph of the group of signed permutations using prefix reversals as generators. By exploring the combinatorial properties and fault-tolerance of [Formula: see text], we show [Formula: see text] and [Formula: see text] for [Formula: see text]. Moreover, we determine that [Formula: see text] for [Formula: see text].

Characterization of a Class of Generalised Core-satellite Graphs Using Average Degree

Network equilibrium models are significantly distinct in supply chain networks, traffic networks, and e-waste flow networks. The idea of network equilibrium is strongly perceived while determining the tuner sets of a graph (network). Tuner sets are subsets of vertices of the graph G whose degrees are lower than the average degree of G, d(G) that can compensate or balance the presence of vertices whose degrees are greater than d(G). Generalised core-satellite graph <img src=image/13428150_01.gif> comprises <img src=image/13428150_02.gif> copies of <img src=image/13428150_03.gif> (the satellites) meeting in K_c (the core) and it belongs to the family of graphs of diameter two. It has a central core of vertices connected to a few satellites, where all satellite cliques need not be identical and can be of different sizes. Properties like hierarchical structure of large real-world networks, are competently modeled using core-satellite graphs [1, 2, 5]. This family of graphs exhibits the properties similar to scale-free network as they possess anomalous vertex connectivity, where a small fraction of vertices (the core) are densely connected. Since these graphs possess such a structural property, interesting results are obtained for these graphs when tuner sets are determined. In this paper, we have considered the graph <img src=image/13428150_04.gif>, with p > q, a subclass of the generalized core-satellite graph which is a join of η copies of the clique K_q and γ copies of the clique K_p with the core K₁. We have obtained the tuner set for this subclass and established the relation between the Top T(G) and the cardinality of the tuner set <img src=image/13428150_05.gif> through necessary and sufficient conditions. We analyze and characterize these graphs and obtain some interesting results while simultaneously examining the existence of tuner sets.

On the cozero-divisor graphs associated to rings

Let R be a ring with unity. The cozero-divisor graph of a ring R, denoted by is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if and In this paper, first we study the Laplacian spectrum of We show that the graph is Laplacian integral. Further, we obtain the Laplacian spectrum of for where and p, q are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of we characterized the values of n for which the Laplacian spectral radius is equal to the order of Moreover, the values of n for which the algebraic connectivity and vertex connectivity of coincide are also described. At the final part of this paper, we obtain the Wiener index of for arbitrary n.

Rainbow Vertex Connection Number pada Keluarga Graf Roda

The rainbow vertex connection was first introduced by krivelevich and yuster in 2009 which is an extension of the rainbow connection. Let graph $G =(V,E)$ is a connected graph. Rainbow vertex-connection is the assignment of color to the vertices of a graph $G$, if every vertex on graph $G$ is connected by a path that has interior vertices with different colors. The minimum number of colors from the rainbow vertex coloring in graph $G$ is called rainbow vertex connection number which is denoted $rvc(G)$. The result of the research are the rainbow vertex connection number of family wheel graphs.

Analisis rainbow vertex connection pada beberapa graf khusus dan operasinya

The vertex colored graph G is said rainbow vertex cennected, if for every two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex connected. On this research, will be raised the issue of how to produce graphs the results of some special graph and how to find the rainbow vertex connection. Operation that use cartesian product, crown product, and shackle. Theorem in this research rainbow vertex connection number in graph the results of operations Wd3,m □ Pn,,Wd3,m ⵙ Pn, and shack(Btm,v,n).

Improvements on Hippchen's conjecture

Let G be a k-connected graph on n vertices. Hippchen's Conjecture (2008) states that two longest paths in G share at least k vertices. Gutiérrez (2020) recently proved the conjecture when k≤4 or k≥n−23. We improve upon both results; namely, we show that two longest paths in G share at least k vertices when k=5 or k≥n+25. This completely resolves two conjectures by Gutiérrez in the affirmative.

Connectivity of superpower graphs of some non-abelian finite groups

For a finite group [Formula: see text] the superpower graph [Formula: see text] of [Formula: see text] is an undirected simple graph with vertex set [Formula: see text] and two vertices in [Formula: see text] are adjacent in [Formula: see text] if and only if the order of one divides the order of the other in the group [Formula: see text]. In this paper, we give sharp bounds for the vertex connectivity of superpower graphs [Formula: see text] and [Formula: see text] of dihedral group [Formula: see text] and dicyclic group [Formula: see text]

The Component (Edge) Connectivity of Round Matching Composition Networks

The vertex (edge) connectivity has been regularly used to measure the fault tolerance and reliability of interconnection networks, while it has defects in the assumption that all neighbors of one node will fail concurrently. To overcome this deficiency, some new generalizations of traditional connectivity have been suggested to quantize the size or the number of the connected components of the survival graph. The [Formula: see text]-component (edge) connectivity, one generalization of vertex (edge) connectivity, has been proposed to characterize the vulnerability of multiprocessor systems based on the number of components of the survival graph. In this paper, we determine the [Formula: see text]-component (edge) connectivity of a family of networks, called the round matching composition networks [Formula: see text], which are a class of networks composed of [Formula: see text] ([Formula: see text]) clusters with the same order, linked by [Formula: see text] perfect matchings. By exploring the combinatorial properties and fault-tolerance of [Formula: see text], we establish the [Formula: see text]-component (edge) connectivity [Formula: see text] for [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text].

Bond topology of chain, ribbon and tube silicates. Part I. Graph-theory generation of infinite one-dimensional arrangements of (TO4)n- tetrahedra.

Chain, ribbon and tube silicates are based on one-dimensional polymerizations of (TO4)n- tetrahedra, where T = Si4+ plus P5+, V5+, As5+, Al3+, Fe3+ and B3+. Such polymerizations may be represented by infinite graphs (designated chain graphs) in which vertices represent tetrahedra and edges represent linkages between tetrahedra. The valence-sum rule of bond-valence theory limits the maximum degree of any vertex to 4 and the number of edges linking two vertices to 1 (corner-sharing tetrahedra). The unit cell (or repeat unit) of the chain graph generates the chain graph through action of translational symmetry operators. The (infinite) chain graph is converted into a finite graph by wrapping edges that exit the unit cell such that they link to vertices within the unit cell that are translationally equivalent to the vertices to which they link in the chain graph, and the wrapped graph preserves all topological information of the chain graph. A symbolic algebra is developed that represents the degree of each vertex in the wrapped graph. The wrapped graph is represented by its adjacency matrix which is modified to indicate the direction of wrapped edges, up (+c) or down (-c) along the direction of polymerization. The symbolic algebra is used to generate all possible vertex connectivities for graphs with ≤8 vertices. This method of representing chain graphs by finite matrices may now be inverted to generate all non-isomorphic chain graphs with ≤8 vertices for all possible vertex connectivities. MatLabR2019b code is provided for computationally intensive steps of this method and ∼3000 finite graphs (and associated adjacency matrices) and ∼1500 chain graphs are generated.

Image Representation on Curved Optimal Triangulation

AbstractImage triangulation aims to generate an optimal partition with triangular elements to represent the given image. One bottleneck in ensuring approximation quality between the original image and a piecewise approximation over the triangulation is the inaccurate alignment of straight edges to the curved features. In this paper, we propose a novel variational method called curved optimal triangulation, where not all edges are straight segments, but may also be quadratic Bézier curves. The energy function is defined as the total approximation error determined by vertex locations, connectivity and bending of edges. The gradient formulas of this function are derived explicitly in closed form to optimize the energy function efficiently. We test our method on several models to demonstrate its efficacy and ability in preserving features. We also explore its applications in the automatic generation of stylization and Lowpoly images. With the same number of vertices, our curved optimal triangulation method generates more accurate and visually pleasing results compared with previous methods that only use straight segments.