Disconnected Graph Research Articles

On cyclically 4-connected cubic graphs

A 3-connected cubic graph is cyclically 4-connected if it has at least n ≥ 8 vertices and when removal of a set of three edges results in a disconnected graph, only one component has cycles. By introducing the notion of cycle spread to quantify the distance between pairs of edges, we get a new characterization of cyclically 4-connected graphs. Let Q n and V n denote the ladder and Möbius ladder on n ≥ 8 vertices, respectively. We prove that a 3-connected cubic graph G is cyclically 4-connected if and only if G is either the Petersen graph, Q n or V n for n ≥ 8 , or G is obtained from Q 8 or Q 10 by bridging pairs of edges with cycle spread at least ( 1 , 2 ) . The concept of cycle spread also naturally leads to methods for constructing cyclically k-connected cubic graphs from smaller ones, but for k ≥ 5 the method is not exhaustive.

Heuristic Expanding Disconnected Graph: A Rapid Path Planning Method for Mobile Robots

Existing mobile robots mostly use graph search algorithms for path planning, which suffer from relatively low planning efficiency owing to high redundancy and large computational complexity. Due to the limitations of the neighborhood search strategy, the robots could hardly obtain the most optimal global path. A global path planning algorithm, denoted as EDG*, is proposed by expanding nodes using a well-designed expanding disconnected graph operator (EDG) in this paper. Firstly, all obstacles are marked and their corners are located through the map pre-processing. Then, the EDG operator is designed to find points in non-obstruction areas to complete the rapid expansion of disconnected nodes. Finally, the EDG* heuristic iterative algorithm is proposed. It selects the candidate node through a specific valuation function and realizes the node expansion while avoiding collision with a minimum offset. Path planning experiments were conducted in a typical indoor environment and on the public dataset CSM. The result shows that the proposed EDG* reduced the planning time by more than 90% and total length of paths reduced by more than 4.6%. Compared to A*, Dijkstra and JPS, EDG* does not show an exponential explosion effect in map size. The EDG* showed better performance in terms of path smoothness, and collision avoidance. This shows that the EDG* algorithm proposed in this paper can improve the efficiency of path planning and enhance path quality.

Fixed Point Theorems for Non-self Mappings Using Disconnected Graphs

Objectives: To prove the fixed point theorems for non-self mappings using disconnected graphs. Method: Graph theoretical approach is adopted to prove the fixed point theorems for non-self mappings. In all the previous works, connected graphs were used for establishing the results, but it is demonstrated in this work that disconnected graphs are best suited, and this new approach simplifies the proofs to a greater extent. Findings: The fixed point theorems by Banach, Kannan, Chatterjea, and Bianchini are proved using the new methodology. Novelty: An important part of the results concerning fixed point theorems is proving the iterated sequence to be a Cauchy sequence, and this is amalgamated with the edge sequence of the disconnected graph. Subject Classification: 54H25, 47H10 Keywords: Non-self mapping, Iterated sequence, Disconnected graph, Edge sequence, Fixed point

Nordhaus-Gaddum Type Inequalities for the $k$th Largest Laplacian Eigenvalues

Let $G$ be a simple connected graph and $\mu_1(G) \geq \mu_2(G) \geq \cdots \geq \mu_n(G)$ be the Laplacian eigenvalues of $G$. Let $\overline{G}$ be the complement of $G$. Einollahzadeh et al.[J. Combin. Theory Ser. B, 151(2021), 235–249] proved that $\mu_{n-1}(G)+\mu_{n-1}(\overline{G})\geq 1$. Grijò et al. [Discrete Appl. Math., 267(2019), 176–183] conjectured that $\mu_{n-2}(G)+\mu_{n-2}(\overline{G})\geq 2$ for any graph and proved it to be true for some graphs. In this paper, we prove $\mu_{n-2}(G)+\mu_{n-2}(\overline{G})\geq 2$ is true for some new graphs. Furthermore, we propose a more general conjecture that $\mu_k(G)+\mu_k(\overline{G})\geq n-k$ holds for any graph $G$, with equality if and only if $G$ or $\overline{G}$ is isomorphic to $K_{n-k}\vee H$, where $H$ is a disconnected graph on $k$ vertices and has at least $n-k+1$ connected components. And we prove that it is true for $k\leq \frac{n+1}{2}$, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when $n\geq 2c+8$.

Wiener index of an ideal-based zero-divisor graph of commutative ring with unity

The Wiener index of a connected graph G is W ( G ) = ∑ { u , v } ⊆ V ( G ) d G ( u , v ) . In this paper, we obtain the Wiener index of H-generalized join of graphs G 1 , G 2 , … , G k . As a consequence, we obtain some earlier known results in [Alaeiyan et al. in Aust. J. Basic Appl. Sci. (2011) 5(12): 145–152; Yeh et al. in Discrete Math. (1994) 135: 359–365] and we also obtain the Wiener index of the generalized corona product of graphs. We further show that the ideal-based zero-divisor graph Γ I ( R ) is a H-generalized join of complete graphs and totally disconnected graphs. As a result, we find the Wiener index of the ideal-based zero-divisor graph Γ I ( R ) and we deduce some of the main results in [Selvakumar et al. in Discrete Appl. Math. (2022) 311: 72–84]. Moreover, we show that W ( Γ I ( Z n ) ) is a quadratic polynomial in n, where Z n is the ring of integers modulo n and we calculate the exact value of the Wiener index of Γ Nil ( R ) ( R ) , where Nil(R) is nilradical of R. Furthermore, we give a Python program for computing the Wiener index of Γ I ( Z n ) if I is an ideal of Z n generated by pr , where pr is a proper divisor of n, p is a prime number and r is a positive integer with r ≥ 2 .

Insights from molecular dynamics into the chemistry-structure relationships of calcium aluminosilicate glasses

Calcium aluminosilicate (CAS) glasses are ubiquitous in nature and play an important role in diverse technological applications ranging from structural glasses to sustainable cementitious materials. Understanding the relationship between the chemical composition and the structure of CAS glasses is an essential step towards optimizing their properties for different uses in the future. Here, we use extensive molecular dynamics (MD) simulations to characterize the multiscale structure of CAS glasses over the full compositional range. Analysis of the short and medium range order of the glasses reveals that Lowenstein's rule is widely violated, that silica is more susceptible than alumina to the depolymerizing effects of calcium, and that high-silica glasses favor calcium in lower oxygen coordination states, while the opposite is true for high-alumina ones. We also find the presence of highly coordinated aluminum and tricluster oxygens in high-alumina glasses, which form as a charge compensation mechanism. We find that current theoretical models used to predict oxygen species, oxygen bridge types, or tetrahedral coordination, while overall qualitatively reasonable, simplify the complex interplay between the different oxides which results in inaccurate predictions, particularly for glasses in intermediate compositional regions. Our analysis of the cluster, chain, and ring topological structures in the aluminosilicate network reveals a sharp transition from a connected to a disconnected graph which depends not only on the calcium content of the glass, but also on the ratio of silica to alumina. Glasses in the compositional region corresponding to such transition display the largest ring and longest chain structures of any glass studied.

Orthogonal Inner Product Graphs over Finite Fields of Odd Characteristic

Let F q be a finite field of odd characteristic and 2 ν + δ ≥ 2 be an integer with δ = 0,1 , or 2. The orthogonal inner product graph O i 2 ν + δ , q over F q is defined, and a class of subgroup of the automorphism groups of O i 2 ν + δ , q is determined. We show that O i 2 ν + δ , q is a disconnected graph if 2 ν + δ = 2 ; otherwise, it is not. Moreover, we give necessary and sufficient conditions for two vertices and two edges of O i 2 ν + δ , q , respectively, which are in the same orbit under the action of a subgroup of the automorphism group of O i 2 ν + δ , q .

Disconnected matchings

In 2005, Goddard et al. (2005) [12] asked the computational complexity of determining the maximum cardinality of a matching whose vertex set induces a disconnected graph. In this paper, we answer this question. In fact, we consider the generalized problem of finding c-disconnected matchings; such matchings are ones whose vertex sets induce subgraphs with at least c connected components. We show that, for every fixed c≥2, this problem is NP-complete even if we restrict the input to bounded diameter bipartite graphs, while can be solved in polynomial time if c=1. For the case when c is part of the input, we show that the problem is NP-complete for chordal graphs, while being solvable in polynomial time for interval graphs. Finally, we explore the parameterized complexity of the problem. We present an FPT algorithm under the treewidth parameterization, and an XP algorithm for graphs with a polynomial number of minimal separators when parameterized by c. We complement these results by showing that, unless NP⊆coNP/poly, the related Induced Matching problem does not admit a polynomial kernel when parameterized by vertex cover and size of the matching nor when parameterized by vertex deletion distance to clique and size of the matching. As for Connected Matching, we show how to obtain a maximum connected matching in linear time given an arbitrary maximum matching in the input.

Relay node placement in wireless sensor network for manufacturing industry

Relay nodes are necessary to maintain scalability and increase longevity as the number of manufacturing industrial sensors grows. In a fixed-budget circumstance, however, the cost of purchasing the bare minimum of relay nodes to connect the network may exceed the budget. Although it is hard to establish a network that connects all sensor nodes, in this case, a network with a high level of connection is still desirable. This paper proposes two metrics for determining the connectedness of a disconnected graph of sensor nodes and determining the optimum deployment method for relay nodes in a network with the highest connectedness while staying within a budget restriction. The metrics are the number of connected graph components and the size of the most significant connected graph component. Prim's algorithm and the approximation minimum spanning tree algorithm are applied to construct a disconnected graph and discover the best relay node placement to solve these two criteria. Compared to the other metrics, simulation findings suggest that prioritizing the most significant connected components in the disconnected graph can yield superior outcomes by deploying the fewest number of relay nodes while retaining the connectedness of the graph.

The Floyd‐Warshall all‐pairs shortest paths algorithm for disconnected and very sparse graphs

AbstractThe Floyd‐Warshall algorithm is the most popular algorithm for determining the shortest paths between all vertex pairs in a graph. It is a very simple and an elegant algorithm. However, for graphs without any negative weighted edges, using Dijkstra's shortest path algorithm for every vertex as a source vertex to produce all‐pairs shortest paths works significantly better than the Floyd‐Warshall algorithm, especially for large graphs. Furthermore, for graphs with negative weighted edges, with no negative cycle, in general Johnson's algorithm also performs better than the Floyd‐Warshall algorithm for large graphs. Johnson's algorithm first transforms the graph into a non‐negative one by using the Bellman‐Ford algorithm, then applies the Dijkstra's algorithm to the transformed graph. Thus, mainly, the Floyd‐Warshall algorithm is quite inefficient, especially for large graphs. In this paper, we show a simple improvement on the Floyd‐Warshall algorithm that will increases its efficiency, especially for very sparse graphs (i.e., the number of its edges is less than the number of its vertices), so it can be used instead of more complicated alternatives. We also show that our approach is also very effective for denser disconnected graphs. Since the new algorithm modifies the original Floyd‐Warshall algorithm, it is mainly aimed for directed graphs without negative cycles. Most programmers prefer to implement the Floyd‐Warshall algorithm over more complicated but more efficient alternatives for solving all‐pairs shortest path problems. In this work, we show that without the addition of any complicated data structures, the performance of the Floyd‐Warshall algorithm can be improved very easily. Our practical approach works even better than its alternatives for large sparse graphs.

Component (edge) connectivity of pancake graphs

The l-component (edge) connectivity of a graph G, denoted by cκl(G) (cλl(G)), is the minimum number of vertices (edges) whose removal from G results in a disconnected graph with at least l components. The pancake graph Pn is a popular underlying topology for distributed systems. In the paper, we determine the cκl(Pn) and cλl(Pn) for 3 ≤ l ≤ 5.

3D Multi-Object Tracking Using Graph Neural Networks With Cross-Edge Modality Attention

Online 3D multi-object tracking (MOT) has witnessed significant research interest in recent years, largely driven by demand from the autonomous systems community. However, 3D offline MOT is relatively less explored. Labeling 3D trajectory scene data at a large scale while not relying on high-cost human experts is still an open research question. In this work, we propose Batch3DMOT which follows the tracking-by-detection paradigm and represents real-world scenes as directed, acyclic, and category-disjoint tracking graphs that are attributed using various modalities such as camera, LiDAR, and radar. We present a multi-modal graph neural network that uses a cross-edge attention mechanism mitigating modality intermittence, which translates into sparsity in the graph domain. Additionally, we present attention-weighted convolutions over frame-wise k-NN neighborhoods as suitable means to allow information exchange across disconnected graph components. We evaluate our approach using various sensor modalities and model configurations on the challenging nuScenes and KITTI datasets. Extensive experiments demonstrate that our proposed approach yields an overall improvement of 3.3% in the AMOTA score on nuScenes thereby setting the new state-of-the-art for 3D tracking and further enhancing false positive filtering.

Component Connectivity of Alternating Group Networks and Godan Graphs

Connectivity is an important index to evaluate the reliability and fault tolerance of a graph. As a natural extension of the connectivity of graphs, the [Formula: see text]-component connectivity of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of vertices whose removal from [Formula: see text] results in a disconnected graph with at least [Formula: see text] components. It is a scientific issue to determine the exact values of [Formula: see text] for distinguishing the fault tolerability of networks. However, [Formula: see text]-component connectivity of many well-known interconnection networks has not been explored even for small [Formula: see text]. For the [Formula: see text]-dimensional alternating group networks [Formula: see text] and [Formula: see text]-dimensional godan graphs [Formula: see text], we show that [Formula: see text] for [Formula: see text], and [Formula: see text] for [Formula: see text] and [Formula: see text].

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].

Annihilator graphs of a commutative semigroup whose Zero-divisor graphs are a complete graph with an end vertex

Abstract Suppose that the zero-divisor graph of a commutative semi-group S, be a complete graph with an end vertex. In this paper, we determine the structure of the annihilator graph S and we show that if Z(S)= S, then the annihilator graph S is a disconnected graph.

The Locating-chromatic Number of Disjoint Union of Fan Graphs

Let G = (V,E) is a connected graph and c is a k-coloring of G. The color class of G is the set of colored vertexs i, denoted by Ci for 1 <= i <= k. Let phi is a ordered partition from V (G) to independent color classes that is C1;C2; ...;Ck, with vertexs of Ci given color by i, 1 <= i <= k. Distance of a vertex v in V to Ci denoted by d(v,Ci) is min {d(v, x)|x in Ci}. The color codes of a vertex v in V is the ordered k-vector c(Phi|v) = (d(v,C1), d(v,C2), ..., d(v,Ck)) where d(v,Ci) = min {d(v, x | x in Ci)} for 1 <= i <= k. If distinct vertices have distinct color codes, then c is called a locating-coloring of G. The locating-chromatic numberXL(G) is the minimum number of colors in a locating-coloring of G. Let H is a disconnected graph and c is a k-coloring of H then induced partition of Phi from V(H). The coloring c is locating k-coloring of H if all vertices of H have distinct color codes. The locating-chromatic number of H, denoted by XL'(H), is the smallest k such that H admits a locating-coloring with k colors. In this paper, we study the locating-chromatic number of disjoint union of fan graphs.

Enumerate the Number of Vertices Labeled Connected Graph of Order Seven Containing No Parallel Edges

A graph that is connected G(V,E) is a graph in which there is at least one path connecting every two vertices in G; otherwise, it is called a disconnected graph. Labels or values can be assigned to the vertices or edges of a graph. A vertex-labeled graph is one in which only the vertices are labeled, and an edges-labeled graph is one in which only edges are assigned values or labels. If both vertices and edges are labeled, the graph is referred to as total labeling. If given n vertices and m edges, numerous graphs can be made, either connected or disconnected. This study will be discussed the number of disconnected vertices labeled graphs of order seven containing no parallel edges and may contain loops. The results show that number of vertices labeled connected graph of order seven with no parallel edges is N(G7,m, g)l= 6,727×Cm6; while for 7≤g≤ 21, N(G7,m, g)l= kg C(m−(g−6))g−1, where k7 =30,160, k8 = 30,765, k9=21,000, k10 =28,364, k11= 26,880, k12=26,460 , k13 = 20,790, k14 =10,290, k15 = 8,022, k16 = 2,940, k17 =4,417, k18 = 2,835, k19 =210, k20 = 21, k21= 1.

The $$\mathrm {v}$$-number of monomial ideals

We show that the \(\mathrm {v}\)-number of an arbitrary monomial ideal is bounded below by the \(\mathrm {v}\)-number of its polarization and also find a criteria for the equality. By showing the additivity of associated primes of monomial ideals, we obtain the additivity of the v-numbers for arbitrary monomial ideals. We prove that the \(\mathrm {v}\)-number \(\mathrm {v}(I(G))\) of the edge ideal I(G), the induced matching number \(\mathrm {im}(G)\) and the regularity \(\mathrm {reg}(R/I(G))\) of a graph G, satisfy \(\mathrm {v}(I(G))\le \mathrm {im}(G)\le \mathrm {reg}(R/I(G))\), where G is either a bipartite graph, or a \((C_{4},C_{5})\)-free vertex decomposable graph, or a whisker graph. There is an open problem in Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021), whether \(\mathrm {v}(I)\le \mathrm {reg}(R/I)+1\), for any square-free monomial ideal I. We show that \(\mathrm {v}(I(G))>\mathrm {reg}(R/I(G))+1\), for a disconnected graph G. We derive some inequalities of \(\mathrm {v}\)-numbers which may be helpful to answer the above problem for the case of connected graphs. We connect \(\mathrm {v}(I(G))\) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that \(\mathrm {reg}(R/I(G))\) can be arbitrarily larger than \(\mathrm {v}(I(G))\). Also, we try to see how the \(\mathrm {v}\)-number is related to the Cohen–Macaulay property of square-free monomial ideals.

Vulnerability analysis of multiprocessor system based on burnt pancake networks

With the large scale network’s popularity in different fields, such as cloud computing, privacy protection, social networks and so on, the robustness analysis of networks against faulty processors has recently attracted more attention. When some processors are attacked and lose function, how to characterize the communication ability and efficiency of the surviving network is very crucial. From the viewpoint of graph theory, the size of the maximal component after removing certain faulty vertices can be viewed as a metric of fault tolerability, which is reckoned as an important dimension for the robustness analysis of a network. The h-component connectivity of a network G, denoted by cκh(G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least h components. In this paper, we show that when removing any subset X with order of at most 5n−9 in the burnt pancake network BPn(n≥5), BPn∖X has the largest component C with |C|≥|V(BPn)|−|X|−4 vertices, and all small components present the following structures: (a) an empty graph; (b) one 3-path, one K1,3, one K1,2, or an edge, or an isolated vertex; (c) one K1,2 and an edge, one K1,2 and an isolated vertex, two edges, one edge and an isolated vertex, or two isolated vertices; (d) an edge and two isolated vertices, or three isolated vertices; (e) four isolated vertices. Besides, we determine that the 6-component connectivity of BPn is cκ6(BPn)=5n−6 for n≥5.

Local limit theorems for subgraph counts

We introduce a general framework for studying anticoncentration and local limit theorems for random variables, including graph statistics. Our methods involve an interplay between Fourier analysis, decoupling, hypercontractivity of Boolean functions, and transference between ``fixed-size'' and ``independent'' models. We also adapt a notion of ``graph factors'' due to Janson. As a consequence, we derive a local central limit theorem for connected subgraph counts in the Erd\H{o}s-Renyi random graph $G(n,p)$, building on work of Gilmer and Kopparty and of Berkowitz. These results improve an anticoncentration result of Fox, Kwan, and Sauermann and partially answers a question of Fox, Kwan, and Sauermann. We also derive a local limit central limit theorem for induced subgraph counts, as long as $p$ is bounded away from a set of ``problematic'' densities, partially answering a question of Fox, Kwan, and Sauermann. We then prove these restrictions are necessary by exhibiting a disconnected graph for which anticoncentration for subgraph counts at the optimal scale fails for all constant $p$, and finding a graph $H$ for which anticoncentration for induced subgraph counts fails in $G(n,1/2)$. These counterexamples resolve anticoncentration conjectures of Fox, Kwan, and Sauermann in the negative. Finally, we also examine the behavior of counts of $k$-term arithmetic progressions in subsets of $\mathbb{Z}/n\mathbb{Z}$ and deduce a local limit theorem wherein the behavior is Gaussian at a global scale but has nontrivial local oscillations (according to a Ramanujan theta function). These results improve on results of and answer questions of the authors and Berkowitz, and answer a question of Fox, Kwan, and Sauermann.