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The security of a network is closely related to the structure of the network graph. The denser the network graph structure is, the better it can resist attacks. Toughness and isolated toughness are used to characterize the vulnerable programs of the network which have been paid attention from mathematics and computer scholars. On this basis, considering the particularity of the sun component structures, sun toughness was introduced in mathematics and applied to computer networks. From the perspective of modern graph theory, this paper presents the sun toughness conditions of the path factor uniform graph and the path factor critical avoidable graph in P ≥ 2 -factor and P ≥ 3 -factor settings. Furthermore, examples show that the given boundaries are sharp.

In this paper, we study the chemical characteristics of some important molecular structures from the mathematical perspective. By means of graph theory approaches, the Nordhaus–Gaddum-type inequalities for some distance-based indices are presented.

Graph theory plays a vital role in modeling and designing any chemical structure or chemical network. Chemical graph theory helps to understand the molecular structural properties of a molecular graph. The molecular graph is a graph consisting of atoms called vertices and the chemical bond between atoms called edges. In this paper, we have computed the various types of degree-based fifth M-Zagreb indices, general fifth M-Zagreb indices, fifth hyper-M-Zagreb indices, general fifth M1-Zagreb polynomial, general fifth M2-Zagreb polynomial, fifth M1-Zagreb polynomial and fifth M2-Zagreb polynomial, fifth hyper-M1-Zagreb polynomial, fifth hyper-M2-Zagreb polynomial and fifth M3-Zagreb index and their polynomials of a molecular graph namely carbon graphite denoted by CG(m,n) for t-levels.

In this paper, a new fully distributed state estimation (DSE) based on weighted least square (WLS) method and graph theory is proposed for power system. The proposed method is fully distributed so that the centralized facilities, e.g., supervisory control and data acquisition (SCADA) and centralized estimators, are not required. Also, different from the existing DSE methods, the proposed method is a bus-level DSE method, in which the power system is not required to be partitioned into several areas. In order to realize the proposed fully distributed DSE method, a novel information propagation algorithm is developed in this paper. This algorithm has great potential in future applications since it is useful to broadcast the local information of the nodes to the entire system in a fully distributed network. The proposed DSE method is compared with the conventional centralized state estimation method and existing multi-area DSE method in different models in this paper. The results show that the proposed method has better performance than the traditional methods.

Chemical graph theory helps to understand the structural properties of a molecular graph. The molecular graphs are the graphs that consist of atoms called vertices and the covalent bond bet...

The vulnerability of the network is one of the core contents of network security research, and there are several variables to test the network vulnerability from the perspective of network structure. For example, toughness, isolated toughness and their variants are defined from graph theory point of view, and these parameters works during the design phase to test the firmness of network. This paper studies the relationship between the existence of path factor and some variants which connect with vulnerability of networks. Moreover, we show that some of bounds of these variants are sharp.

In the network security research, the vulnerability of the network is always considered in the network designing stage, and there are some variables defined to measure the vulnerability from the perspective of graph theory. The most famous parameters are toughness and isolated toughness, which are also related to the existence of factional factor and path factor. In this article, we mainly study the relationship between the existence of matching extendable and some variants which connect with vulnerability of network. By means of graph theory tricks, several results are given from a theoretical perspective.

Sierpinski graphs constitute an extensively studied class of graphs of fractal nature applicable in topology, mathematics of Tower of Hanoi, computer science, and elsewhere. A large number of properties like physico-chemical properties, thermodynamic properties, chemical activity, biological activity, etc. are determined by the chemical applications of graph theory. These properties can be characterized by certain graph invariants referred to as topological indices. In QRAR/QSPR study these graph invariants has played a vital role. In this paper, we study the molecular topological properties of Sierpinski networks and derive the analytical closed formulas for the atom-bond connectivity (ABC) index, geometric-arithmetic (GA) index, and fourth and fifth version of these topological indices for Sierpinski networks denoted by S(n, k).

This paper proposes a new polynomial time constant factor approximation algorithm for a more-a-decade-long open NP-hard problem, the minimum four-connected $m$ -dominating set problem in unit disk graph (UDG) with any positive integer $m \geq 1$ for the first time in the literature. We observe that it is difficult to modify the existing constant factor approximation algorithm for the minimum three-connected $m$ -dominating set problem to solve the minimum four-connected $m$ -dominating set problem in UDG due to the structural limitation of Tutte decomposition, which is the main graph theory tool used by Wang et al. to design their algorithm. To resolve this issue, we first reinvent a new constant factor approximation algorithm for the minimum three-connected $m$ -dominating set problem in UDG and later use this algorithm to design a new constant factor approximation algorithm for the minimum four-connected $m$ -dominating set problem in UDG.

In QSAR/QSPR study, physico-chemical properties and topological indices such as Randic, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. A topological index is actually designed by transforming a chemical structure into a numeric number. These topological indices correlate certain physico-chemical properties like boiling point, stability, strain energy etc. of chemical compounds. Graph theory has found a considerable use in this area of research. The topological indices of certain interconnection networks were studied recently by Imran et al. (Appl Math Comput 244:936–951, 2014). In this paper, we extend this study to $$n\times n$$ Sudoku graphs and derive analytical closed results of general Randic index $$R_{\alpha }(G)$$ for different values of “ $$\alpha $$ ” for Sudoku (SK). We also compute the general Randic, first Zagreb, ABC, GA, $$ABC_{4}$$ and $$GA_{5}$$ indices and give closed formulae of these indices for Sudoku graphs.

As an effective modeling, analysis and computational tool, graph theory is widely used in biological mathematics to deal with various biology problems. In the field of microbiology, graph can express the molecular structure, where cell, gene or protein can be denoted as a vertex, and the connect element can be regarded as an edge. In this way, the biological activity characteristic can be measured via topological index computing in the corresponding graphs. In our article, we mainly study the biology features of biological networks in terms of eccentric topological indices computation. By means of graph structure analysis and distance calculating, the exact expression of several important eccentric related indices of hypertree network and X-tree are determined. The conclusions we get in this paper illustrate that the bioengineering has the promising application prospects. Keywords: Biological mathematics, DNA sequence, Biological networks, Topological index

Graph theory is used for modeling, designing, analysis and understanding chemical structures or chemical networks and their properties. The molecular graph is a graph consisting of atoms called vertices and the chemical bond between atoms called edges. In this article, we study the chemical graphs of carbon graphite and crystal structure of cubic carbon. Moreover, we compute and give closed formulas of degree based additive topological indices, namely hyper-Zagreb index, first multiple and second multiple Zagreb indices, and first and second Zagreb polynomials.

Graph theory has much advancement in the field of mathematical chemistry. Recently, chemical graph theory has become very popular among researchers because of its wide applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields.In this article, we study the chemical graph of copper oxide and compute degree based topological indices mainly ABC, GA, ABC4, GA5, general Randić index and Zagreb index for copper(II) oxide, CuO. Furthermore, we give exact formulas of these indices which are helpful in studying the underlying topologies.

As an efficient theoretical tool, graph theory is widely used in computing chemistry. In terms of index computation on molecular graphs, the researchers can learn the potential properties of chemical compounds, including drugs, materials, and organics. In this paper, by means of distance computation, we study the eccentric version indices of cycloalkanes which occur quite frequently in the chemical drugs and other compounds. The promising prospects of the application for the physical, chemical, medical, and pharmacy engineering are illustrated by theoretical conclusions obtained in this article.

Resting state functional magnetic resonance imaging (rsfMRI) has emerged as a versatile tool for non-invasive measurement of functional connectivity patterns in the brain. RsfMRI brain dynamics in rodents, non-human primates, and humans share similar properties; however, little is known about the resting state functional connectivity patterns in the ferret, an animal model with high potential for developmental and cognitive translational study. To address this knowledge-gap, we performed rsfMRI on anesthetized ferrets using a 9.4 T MRI scanner, and subsequently performed group-level independent component analysis (gICA) to identify functionally connected brain networks. Group-level ICA analysis revealed distributed sensory, motor, and higher-order networks in the ferret brain. Subsequent connectivity analysis showed interconnected higher-order networks that constituted a putative default mode network (DMN), a network that exhibits altered connectivity in neuropsychiatric disorders. Finally, we assessed ferret brain topological efficiency using graph theory analysis and found that the ferret brain exhibits small-world properties. Overall, these results provide additional evidence for pan-species resting-state networks, further supporting ferret-based studies of sensory and cognitive function.

The problem of data transmission in communication network can betransformed into the problem of fractional factor existing in graph theory. Inrecent years, the data transmission problem in the specificnetwork conditions has received a great deal of attention, and itraises new demands to the corresponding mathematical model. Underthis background, many advanced results are presented on fractionalcritical deleted graphs and fractional ID deleted graphs. In thispaper, we determine that $G$ is a fractional \begin{document}$ (g,f,n',m) $\end{document} -critical deleted graph if \begin{document}$ δ(G)≥\frac{b^{2}(i-1)}{a}+n'+2m $\end{document} , \begin{document}$ n>\frac{(a+b)(i(a+b)+2m-2)+bn'}{a} $\end{document} , and \begin{document}$|N_{G}(x_{1})\cup N_{G}(x_{2})\cup···\cup N_{G}(x_{i})|≥\frac{b(n+n')}{a+b}$ \end{document} for any independent subset \begin{document}$ \{x_{1},x_{2},..., x_{i}\} $\end{document} of \begin{document}$ V(G) $\end{document} . Furthermore, the independent set neighborhood union condition for a graph to be fractional ID- \begin{document}$ (g,f,m) $\end{document} -deleted is raised. Some examples will be manifested to show the sharpness of independent set neighborhood union conditions.

In this paper, we study the problem of computing quality fault-tolerant virtual backbone in homogeneous wireless network, which is defined as the k-connected m-dominating set problem in a unit disk graph. This problem is NP-hard, and thus many efforts have been made to find a constant factor approximation algorithm for it, but never succeeded so far with arbitrary k ≥ 3 and m ≥ 1 pair. We propose a new strategy for computing a smaller-size 3-connected m-dominating set in a unit disk graph with any m ≥ 1. We show the approximation ratio of our algorithm is constant and its running time is polynomial. We also conduct a simulation to examine the average performance of our algorithm. Our result implies that while there exists a constant factor approximation algorithm for the k-connected m-dominating set problem with arbitrary k ≤ 3 and m ≥ 1 pair, the k-connected m-dominating set problem is still open with k > 3.

Constructing a reliable data term and occlusion handling are two important issues for energy model based stereo method. In this paper, we at first use a 2-step adaptive support-weight correlation approach to get a reliable correlation volume. Then a pixel classification is proposed which classifies pixels into three classes: occluded, unstable and stable. For each pixel, according its class, a confidence weight is assigned. After that a new energy model is then constructed by integrating the correlation volume and the confidence weight. Finally, through minimizing this energy using Graph cuts, a better disparity map is obtained. Experimental results on the Middlebury data set show that our proposed method has the similar good performance with the top rank Graph Cuts based algorithms listed on the Middlebury homepage.

Document clustering techniques mostly depend on models that impose explicit and/or implicit priori assumptions as to the number, size, disjunction characteristics of clusters, and/or the probability distribution of clustered data. As a result, the clustering effects tend to be unnatural and stray away more or less from the intrinsic grouping nature among the documents in a corpus. We propose a novel graph-theoretic technique called Clique Percolation Clustering (CPC). It models clustering as a process of enumerating adjacent maximal cliques in a random graph that unveils inherent structure of the underlying data, in which we unleash the commonly practiced constraints in order to discover natural overlapping clusters. Experiments show that CPC can outperform some typical algorithms on benchmark data sets, and shed light on natural document clustering.