Disconnected Graph Research Articles

The fault tolerance of [formula omitted]-bubble-sort networks

The h-good neighbor connectivity κh(G) is an important parameter to evaluate the fault tolerance of an interconnection network G. So far, almost all known results of κh(G) are about small h′s except the hypercubes, the star graphs, the k-ary n-cubes and hierarchical hypercube network HHCn et al. In this paper, we focus on κh(Bn,k) for the (n,k)-bubble-sort network Bn,k, which is a generalization of the bubble-sort graph Bn as it is a special case of (n,k)-bubble-sort networks for k=n−1. We show that κh(Bn,k)=n+h(k−2)−1 for 2≤k≤⌈n+32⌉ and 0≤h≤⌊n−32⌋, which implies that at least n+h(k−2)−1 vertices of Bn,k have to be deleted to get a disconnected graph without vertices of degree less than h.

Graphs Whose Independence Fractals are Line Segments

Let G be a simple graph. By an independent set in G, we mean a set of pairwise non-adjacent vertices in G. The independence polynomial of G is defined as $$I_G(z)=i_0 + i_1 z + i_2 z^2+\cdots +i_\alpha z^{\alpha }$$ , where $$i_m=i_m(G)$$ is the number of independent sets in G with cardinality m and $$\alpha =\alpha (G)$$ denotes the cardinality of a largest independent set in G (known as the independence number of G). Let $$G^k$$ denote the k-times lexicographic product of G with itself. The set of roots of $$I_{G^k}$$ is known to converge as k tends to $$\infty $$ , with respect to the Hausdorff metric, and the limiting set is known as the independence attractor. The independence fractal of a graph is the limiting set of roots of the reduced independence polynomial $$I_{G^k}-1$$ of $$G^k$$ as k tends to $$\infty $$ . In this article, we consider the independence fractals of graphs with independence number 3. We attempt to find all such graphs whose independence fractal is a line segment. It is shown that the independence fractal and the independence attractor coincide when the earlier is a line segment. The line segment turns out to be an interval $$[-\frac{4}{k}, 0]$$ for $$k \in \{1, 2, 3, 4\}$$ . It is found that each of these graphs have 9 vertices and there are exactly 13 such disconnected graphs. We show that there does not exist any connected graph for $$k=4$$ . For $$k=1$$ , there are 17 such connected graphs and for $$k=2,3$$ the number is quite large.

On Component Connectivity of Hierarchical Star Networks

For an integer [Formula: see text], 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 or a graph with fewer than [Formula: see text] vertices. This naturally generalizes the classical connectivity of graphs defined in term of the minimum vertex-cut. This kind of connectivity can help us to measure the robustness of the graph corresponding to a network. The hierarchical star networks [Formula: see text], proposed by Shi and Srimani, is a new level interconnection network topology, and uses the star graphs as building blocks. In this paper, by exploring the combinatorial properties and fault-tolerance of [Formula: see text], we study the [Formula: see text]-component connectivity of hierarchical star networks [Formula: see text]. We obtain the results: [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text].

On the fold thickness of graphs

The graph G' obtained from a graph G by identifying two nonadjacent vertices in G having at least one common neighbor is called a 1-fold of G. A sequence G_0, G_1, G_2, ldots , G_k of graphs such that G_0=G and G_i is a 1-fold of G_{i-1} for each i=1, 2, ldots , k is called a uniform k-folding of G if the graphs in the sequence are all singular or all nonsingular. The fold thickness of G is the largest k for which there is a uniform k-folding of G. We show here that the fold thickness of a singular bipartite graph of order n is n-3. Furthermore, the fold thickness of a nonsingular bipartite graph is 0, i.e., every 1-fold of a nonsingular bipartite graph is singular. We also determine the fold thickness of some well-known families of graphs such as cycles, fans and some wheels. Moreover, we investigate the fold thickness of graphs obtained by performing operations on these families of graphs. Specifically, we determine the fold thickness of graphs obtained from the cartesian product of two graphs and the fold thickness of a disconnected graph whose components are all isomorphic.

DRACON: disconnected graph neural network for atom mapping in chemical reactions.

Machine learning solved many challenging problems in computer-assisted synthesis prediction (CASP). We formulate a reaction prediction problem in terms of node-classification in a disconnected graph of source molecules and generalize a graph convolution neural network for disconnected graphs. Here we demonstrate that our approach can successfully predict centres of reaction and atoms of the main product. A set of experiments using the USPTO dataset demonstrates excellent performance and interpretability of the proposed model. Implicitly learned latent vector representation of chemical reactions strongly correlates with the class of the chemical reaction. Reactions with similar templates group together in the latent vector space.

Analysis on component connectivity of bubble-sort star graphs and burnt pancake graphs

The ℓ-component connectivity of a graph G, denoted by cκℓ(G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least ℓ components or a graph with fewer than ℓ vertices. This is a natural generalization of the classical connectivity of graphs defined in terms of the minimum vertex-cut. Since this parameter can be used to evaluate the reliability and fault tolerance of a graph G corresponding to a network, determining the exact values of cκℓ(G) is an important issue on the research topic of networks. However, it has been pointed out in Hsu et al. (2012) that determining ℓ-component connectivity is still unsolved in most interconnection networks even for small ℓ’s. Let BSn and BPn denote the n-dimensional bubble-sort star graph and the n-dimensional burnt pancake graph, respectively. In this paper, for BSn, we determine the values: cκ3(BSn)=4n−9 for n⩾3, and cκ4(BSn)=6n−16 and cκ5(BSn)=8n−24 for n⩾4. Similarly, for BPn, we determine the values: cκ3(BPn)=2n−1 and cκ4(BPn)=3n−2 for n⩾4, and cκ5(BPn)=4n−4 for n⩾5.

The locating chromatic number of disconnected graph with path and cycle graph as its components

A special case of the graph partition dimension is The locating-chromatic number of a graph. In 2002, The concept of locating-chromatic number of a connected graph was introduced by Chartrand et al.. In 2014, definition locating-chromatic number for a disconnected graph is extended by Welyyanti et al. As a continuation, we determine the locating-chromatic number of some disconnected graphs with path and cycle as its components.

The Locating Chromatic Number of a Disjoint Union of Some Double Stars

In this paper we discuss the locating chromatic number of a disconnected graph, namely a disjoint union of some double stars. To determine the locating-chromatic number of a graph, first we construct the upper bound of the number and observe the minimum coloring used. Then, to determine the lower bound of the number, we use the trivial lower bound for some leaves which incident to one vertex. Two main theorems about the locating chromatic number of a disjoint union of some double stars. We obtained some original results of the locating chromatic number of disjoint union of some double stars. Moreover, we can generalize the graph such that the locating chromatic number remains the same with the previous one.

Solvable groups whose character degree graphs generalize squares

Abstract Let G be a solvable group, and let Δ ⁢ ( G ) {\Delta(G)} be the character degree graph of G. In this paper, we generalize the definition of a square graph to graphs that are block squares. We show that if G is a solvable group so that Δ ⁢ ( G ) {\Delta(G)} is a block square, then G has at most two normal nonabelian Sylow subgroups. Furthermore, we show that when G is a solvable group that has two normal nonabelian Sylow subgroups and Δ ⁢ ( G ) {\Delta(G)} is block square, then G is a direct product of subgroups having disconnected character degree graphs.

On Computing Component (Edge) Connectivities of Balanced Hypercubes

Abstract For an integer $\ell \geqslant 2$, the $\ell $-component connectivity (resp. $\ell $-component edge connectivity) of a graph $G$, denoted by $\kappa _{\ell }(G)$ (resp. $\lambda _{\ell }(G)$), is the minimum number of vertices (resp. edges) whose removal from $G$ results in a disconnected graph with at least $\ell $ components. The two parameters naturally generalize the classical connectivity and edge connectivity of graphs defined in term of the minimum vertex-cut and the minimum edge-cut, respectively. The two kinds of connectivities can help us to measure the robustness of the graph corresponding to a network. In this paper, by exploring algebraic and combinatorial properties of $n$-dimensional balanced hypercubes $BH_n$, we obtain the $\ell $-component (edge) connectivity $\kappa _{\ell }(BH_n)$ ($\lambda _{\ell }(BH_n)$). For $\ell $-component connectivity, we prove that $\kappa _2(BH_n)=\kappa _3(BH_n)=2n$ for $n\geq 2$, $\kappa _4(BH_n)=\kappa _5(BH_n)=4n-2$ for $n\geq 4$, $\kappa _6(BH_n)=\kappa _7(BH_n)=6n-6$ for $n\geq 5$. For $\ell $-component edge connectivity, we prove that $\lambda _3(BH_n)=4n-1$, $\lambda _4(BH_n)=6n-2$ for $n\geq 2$ and $\lambda _5(BH_n)=8n-4$ for $n\geq 3$. Moreover, we also prove $\lambda _\ell (BH_n)\leq 2n(\ell -1)-2\ell +6$ for $4\leq \ell \leq 2n+3$ and the upper bound of $\lambda _\ell (BH_n)$ we obtained is tight for $\ell =4,5$.

The connected size Ramsey number for matchings versus small disconnected graphs

Let F , G , and H be simple graphs. The notation F → ( G , H ) means that if all the edges of F are arbitrarily colored by red or blue, then there always exists either a red subgraph G or a blue subgraph H . The size Ramsey number of graph G and H , denoted by r ( G , H ) is the smallest integer k such that there is a graph F with k edges satisfying F → ( G , H ). In this research, we will study a modified size Ramsey number, namely the connected size Ramsey number. In this case, we only consider connected graphs F satisfying the above properties. This connected size Ramsey number of G and H is denoted by r c ( G , H ). We will derive an upper bound of r c ( n K 2 , H ), n ≥ 2 where H is 2 P m or 2 K 1, t , and find the exact values of r c ( n K 2 , H ), for some fixed n .

Super Connectivity of Erdős-Rényi Graphs

The super connectivity κ ′ ( G ) of a graph G is the minimum cardinality of vertices, if any, whose deletion results in a disconnected graph that contains no isolated vertex. G is said to be r-super connected if κ ′ ( G ) ≥ r . In this note, we establish some asymptotic almost sure results on r-super connectedness for classical Erdős–Rényi random graphs as the number of nodes tends to infinity. The known results for r-connectedness are extended to r-super connectedness by pairing off vertices and estimating the probability of disconnecting the graph that one gets by identifying the two vertices of each pair.

Classifying character degree graphs with six vertices

We investigate prime character degree graphs of solvable groups that have six vertices. There are 112 non-isomorphic connected graphs with six vertices, of which all except nine are classified in this paper. We also completely classify the disconnected graphs with six vertices.

Measuring the Vulnerability of Alternating Group Graphs and Split-Star Networks in Terms of Component Connectivity

For an integer $\ell\geqslant 2$, the $\ell$-component connectivity of a graph $G$, denoted by $\kappa_{\ell}(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a graph with fewer than $\ell$ vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and is a good measure of robustness for the graph corresponding to a network. So far, the exact values of $\ell$-connectivity are known only for a few classes of networks and small $\ell$'s. It has been pointed out in~[Component connectivity of the hypercubes, Int. J. Comput. Math. 89 (2012) 137--145] that determining $\ell$-connectivity is still unsolved for most interconnection networks, such as alternating group graphs and star graphs. In this paper, by exploring the combinatorial properties and fault-tolerance of the alternating group graphs $AG_n$ and a variation of the star graphs called split-stars $S_n^2$, we study their $\ell$-component connectivities. We obtain the following results: (i) $\kappa_3(AG_n)=4n-10$ and $\kappa_4(AG_n)=6n-16$ for $n\geqslant 4$, and $\kappa_5(AG_n)=8n-24$ for $n\geqslant 5$; (ii) $\kappa_3(S_n^2)=4n-8$, $\kappa_4(S_n^2)=6n-14$, and $\kappa_5(S_n^2)=8n-20$ for $n\geqslant 4$.

Relay node placement under budget constraint

The relay node placement problem in the wireless sensor network domain has been studied extensively. But under a fixed budget, it may be impossible to procure the minimum number of relay nodes needed to design a connected network of sensor and relay nodes. Nevertheless, one would still like to design a network with high level of connectedness, or low disconnectedness. In this paper, we introduce the notion of a measure of the “connectedness” of a disconnected graph. We study a family of problems whose goal is to design a network with “maximal connectedness” subject to a fixed budget constraint.

On the limit value of compactness of some graph classes.

In this paper, we study the limit of compactness which is a graph index originally introduced for measuring structural characteristics of hypermedia. Applying compactness to large scale small-world graphs (Mehler, 2008) observed its limit behaviour to be equal 1. The striking question concerning this finding was whether this limit behaviour resulted from the specifics of small-world graphs or was simply an artefact. In this paper, we determine the necessary and sufficient conditions for any sequence of connected graphs resulting in a limit value of CB = 1 which can be generalized with some consideration for the case of disconnected graph classes (Theorem 3). This result can be applied to many well-known classes of connected graphs. Here, we illustrate it by considering four examples. In fact, our proof-theoretical approach allows for quickly obtaining the limit value of compactness for many graph classes sparing computational costs.

On finite-time and fixed-time consensus algorithms for dynamic networks switching among disconnected digraphs

ABSTRACT This paper aims to analyse the stability of a class of consensus algorithms with finite-time or fixed-time convergence for dynamic networks composed of agents with first-order dynamics. In particular, in the analysed class a single evaluation of a nonlinear function of the consensus error is performed per each node. The classical assumption of switching among connected graphs is dropped here, allowing to represent failures and intermittency in the communications between agents. Thus, conditions to guarantee finite and fixed-time convergence, even while switching among disconnected graphs, are provided. Moreover, the algorithms of the considered class are computationally simpler than previously proposed finite-time consensus algorithms for dynamic networks, which is an essential feature in scenarios with computationally limited nodes and energy efficiency requirements such as in sensor networks. Simulations illustrate the performance of the proposed consensus algorithms. In the presented scenarios, results show that the settling time of the considered algorithms grows slower than other consensus algorithms for dynamic networks as the number of nodes increases.

Multiresolution Network Models

ABSTRACTMany existing statistical and machine learning tools for social network analysis focus on a single level of analysis. Methods designed for clustering optimize a global partition of the graph, whereas projection-based approaches (e.g., the latent space model in the statistics literature) represent in rich detail the roles of individuals. Many pertinent questions in sociology and economics, however, span multiple scales of analysis. Further, many questions involve comparisons across disconnected graphs that will, inevitably be of different sizes, either due to missing data or the inherent heterogeneity in real-world networks. We propose a class of network models that represent network structure on multiple scales and facilitate comparison across graphs with different numbers of individuals. These models differentially invest modeling effort within subgraphs of high density, often termed communities, while maintaining a parsimonious structure between said subgraphs. We show that our model class is projective, highlighting an ongoing discussion in the social network modeling literature on the dependence of inference paradigms on the size of the observed graph. We illustrate the utility of our method using data on household relations from Karnataka, India. Supplementary material for this article is available online.

The 4-component connectivity of alternating group networks

The ℓ-component connectivity (or ℓ-connectivity for short) of a graph G, denoted by κℓ(G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least ℓ components or a graph with fewer than ℓ vertices. This generalization is a natural extension of the classical connectivity defined in term of minimum vertex-cut. As an application, the ℓ-connectivity can be used to assess the vulnerability of a graph corresponding to the underlying topology of an interconnection network, and thus is an important issue for reliability and fault tolerance of the network. So far, only a little knowledge of results have been known on ℓ-connectivity for particular classes of graphs and small ℓ's. In a previous work, we studied the ℓ-connectivity on n-dimensional alternating group networks ANn and obtained the result κ3(ANn)=2n−3 for n⩾4. In this sequel, we continue the work and show that κ4(ANn)=3n−6 for n⩾4.

About Supergraphs. Part I

Summary Drawing a finite graph is usually done by a finite sequence of the following three operations. 1. Draw a vertex of the graph. 2. Draw an edge between two vertices of the graph. 3. Draw an edge starting from a vertex of the graph and immediately draw a vertex at the other end of it. By this procedure any finite graph can be constructed. This property of graphs is so obvious that the author of this article has yet to find a reference where it is mentioned explicitly. In introductionary books (like [10], [5], [9]) as well as in advanced ones (like [4]), after the initial definition of graphs the examples are usually given by graphical representations rather than explicit set theoretic descriptions, assuming a mutual understanding how the representation is to be translated into a description fitting the definition. However, Mizar [2], [3] does not possess this innate ability of humans to translate pictures into graphs. Therefore, if one wants to create graphs in Mizar without directly providing a set theoretic description (i.e. using the createGraph functor), a rigorous approach to the constructing operations is required. In this paper supergraphs are defined as an inverse mode to subgraphs as given in [8]. The three graph construction operations are defined as modes extending Supergraph similar to how the various remove operations were introduced as submodes of Subgraph in [8]. Many theorems are proven that describe how graph properties are transferred to special supergraphs. In particular, to prove that disconnected graphs cannot become connected by adding an edge between two vertices that lie in the same component, the theory of replacing a part of a walk with another walk is introduced in the preliminaries.