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Quantitative structure property relationship (QSPR) is essential in rational drug design by facilitating the prediction, optimization, and prioritization of potential drug candidates based on their chemical structures and properties, optimizing the utilization of computational resources and minimizing costs. Topological indices play a fundamental role in QSPR studies by providing a quantitative representation of molecular structures and facilitating the prediction of various chemical properties and activities. Metastatic skin cancer can be aggressive and potentially fatal, and thus developing effective drugs can improve patient outcomes, including overall survival rates and progression-free survival. This research work focuses on the structural behaviors of medications used in the treatment of skin cancer, including binimetinib, fisetin, encorafenib, picato, fluorouracil, trametinib, vemurafenib, imiquimod, odomzo, vismodegib, dacarbazine, cobimetinib, dabrafenib, sesamol, curcumin, doxorubicin, temozolomide, paclitaxel, itraconazole, and hyaluronan. We have developed QSPR models involving degree and neighborhood degree sum-based indices and conducted a comparative analysis to highlight the efficacy of the models.

Given a hypergraph ${\mathscr{H}}$, we introduce a new class of evaluation toric codes called edge codes derived from ${\mathscr{H}}$. We analyze these codes, focusing on determining their basic parameters. We provide estimations for the minimum distance, particularly in scenarios involving $d$-uniform clutters. Additionally, we demonstrate that these codes exhibit self-orthogonality. Furthermore, we compute the minimum distances of edge codes for all graphs with five vertices.

Let G be a connected bipartite graph with with bipartition (X, Y) such that |X| ≥ |Y| (≥ 2). Put n = |X|, m = |Y| and l = m + n. Suppose that, for all vertices x ∈ X and y ∈ Y, dist(x,y) = 3 implies d(x) + d(y) ≥ n + 1. We show that G contains a cycle of length 2m. We also give an efficient algorithm to obtain such a cycle. The complexity of this algorithm is O(l³). In case m = n, we find a hamiltonian cycle of G. This generalizes a result given in [10].

The non-commuting graph associated to a group has non-central elements of the graph as vertices and two elements $x$ and $y$ do not form an edge if and only if $xy=yx$. In this paper, we consider non-commuting graphs associated to dihedral and semidihedral groups. We investigate their metric properties such as center, periphery, eccentric graph, closure and interior. We also perform various types of metric identifications on these graphs. Moreover, we generate metric and metric-degree polynomials of these graphs.

Networks with distinct topological structures varies in the ability to resist different kinds of attacks. Node protection in terms of node importance is an effective way to ensure the reliable communication of networks. Thus, the protection of influential node detected by node importance evaluation is beneficial to strengthen the invulnerability and robustness of networks. In this work, the network agglomeration method based on node contraction is utilized to evaluate the influence of nodes and identify important nodes in five kinds of interconnection networks. We show that the agglomeration method, which takes both the degree and position into account, is valid and feasible for these five interconnection networks.

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

In this paper we study the model of opinion dynamics with individuals on nodes of a scale-free network, introducing an opinion evolution mechanism and according to the nodes' degrees defining broad-sense-concept leaders on the network. We compare the strength of opinion influence between leaders and followers. With computer simulations of opinion dynamics, we found that the more complex is the scale-free network, the easier for leaders to make their opinion accepted by the masses.

We study virtual path layouts in ATM networks. Packets are routed along virtual paths in the network by maintaining a routing field whose subfields determine intermediate destinations of the packet, i.e., the endpoints of virtual paths on its way to the final destination. Most of the research on virtual path layouts has focused on tradeoffs between load (i.e., the maximum number of virtual paths passing through a link) and the hop number of the layout (i.e., the maximum number of virtual paths needed to travel between any two nodes).

There is however another important limitation on construction of layouts, resulting from technological properties of switches situated at nodes. This bound is the degree of the layout (i.e., the maximum number of virtual paths with a common endpoint). In this paper we study relations between these three parameters of virtual path layouts, for the all-to-all problem. For any integer h, we show tradeoffs between load and degree of h-hop layouts in the ring and in the mesh by establishing upper and lower bounds on these parameters. Our bounds on the degree of an h-hop layout of given load are asymptotically tight and the bounds on the load of an h-hop layout of given degree are asymptotically tight for constant h.

It was generally believed that scale-free networks would be small-world. In this paper, two models, named Model A and Model B, are proposed to show that certain scale-free networks can be linear-world instead of small-world. By linear-world, it means that the average path length L of the network grows linearly with the total number of nodes N, i.e., L~N. Model A generates a deterministic scale-free network with high assortativity and numerical simulations demonstrate that the network is linear-world when it satisfies degree exponent λ>1. Model B constructs a partially deterministic scale-free network, which is connected by identical small scale-free networks following certain rules. Analytical arguments and numerical simulations both yield L~N which suggests that it is also linear-world. It is further discussed in this paper that the network generated by Model Bcould be either correlated or uncorrelated. This suggests that, inconsistent with the results in related works, uncorrelated scale-free networks can also be linear-world.

Terrorist network may be defined as collection of suspected terrorist nodes which may function in disguise towards accomplishing a terrorist activity. They use extensive communication channel for sharing crucial information. Terrorist network analysis is highly efficacious for intelligence analysis and deriving useful conclusions from available data. Computer Science and Network analysis act as pertinent fields for the study and graphical interpretation of these networks. In this paper, we examine the 26/11 Mumbai attack terrorist network dataset and employ the ELECTRE method for identification of key node in the terrorist network. ELECTRE is an effective multi-criteria decision-making model. It provides a framework for structuring a decision problem integrates the quantitative and qualitative factors of the problem and facilitates easy computation. From the 26/11 Mumbai attack dataset of terrorist network, we have determined that out of several terrorists in the network “Wassi” was the momentous and mastermind of all. The proposed work also demonstrates improvement of result in terms of concurrence, generalization accuracy and genuineness. Based on the solution of ELECTRE framework, it is resolved that the obtained (terrorist) nodes will step up the work of law enforcement agencies and enable them to confine their focus on important members of the terrorist network. Identification of key terrorist is highly important for developing long-term strategies to counter forthcoming terrorist attacks. It can be better implemented during the development of smart city especially for India.

Link prediction of complex network intends to estimate the probability of existence of links between two nodes. In order to improve link prediction accuracy and fully exploit the potentialities of nodes, many studies focus more on the influence of degree on nodes but less on the hybrid influence of degree and H-index. The nodes with a larger degree have more neighbors, and the nodes with larger H-index have more neighbors of neighbors. Meanwhile, weak ties consisting of neighbors with a small degree have powerful strength of intermediary ability and a high probability of passing similarity. A novel link prediction model is proposed considering the hybrid influence of degree and H-index and weak ties, which is called Hybrid Weak Influence, marked as HWI. After experimenting with nine real datasets, the results show that this method can significantly improve the link prediction accuracy, compared with the empirical methods: Common Neighbors (CN), Resource-Allocation (RA) and Adamic/Adar (AA). Meanwhile, the computation complexity is less than the long path algorithm of LP, SRW, PCEN.

In this paper we provide a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, providing also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction presented here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism. A brief discussion of a possibly independent derivation made by Maxwell in 1867 completes this reconstruction. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important role in modern mathematical physics, we offer a direct proof of their equivalence. Explicit examples of its interpretation in terms of oriented area are also provided.

Traditional derivations of general relativity (GR) from the graviton degrees of freedom assume spacetime Lorentz covariance as an axiom. In this paper, we survey recent evidence that GR is the unique spatially-covariant effective field theory of the transverse, traceless graviton degrees of freedom. The Lorentz covariance of GR, having not been assumed in our analysis, is thus plausibly interpreted as an accidental or emergent symmetry of the gravitational sector. From this point of view, Lorentz covariance is a necessary feature of low-energy graviton dynamics, not a property of spacetime. This result has revolutionary implications for fundamental physics.

Generalized Hypercube-Connected-Cycles (GHCC), is a challenging interconnection network, proposed earlier in the literature. In this paper, we discuss how some important, useful algorithms, like, matrix transpose, matrix multiplication and sorting can efficiently be implemented on GHCC. Matrix transpose and matrix-by-matrix multiplication of matrices of order n × n, , takes O(l) and time, respectively, on GHCC(l,m), with lm^l processors. Using the same number of processors, a list of m^l numbers can be sorted in O(l²log³ m) time.

In this paper, we introduce detrended cross-correlation analysis (DCCA) method into visibility graph (VG) algorithm and propose VGDCCA method to reflect the time series irreversibility from a new perspective. The validity and reliability of the proposed VGDCCA method is supported by numerical simulations on synthesized short-term correlated chaotic systems and long-term correlated fractal processes, and by the empirical analysis on stock indices and traffic parameter. The VGDCCA planes suggest that autoregressive fractionally integrated moving average (ARFIMA) and fractional Gaussian noise (FGN) series show time reversible behavior, whatever the long-term correlation of the series is or however strong the persistence or anti-persistence of the series is. Meanwhile, Logistic map with $a=3.3\sim 3.6$ and Hénon map show more time irreversible behaviors than those of ARFIMA, FGN series and other Logistic maps. It can be found that the relationship between cross-correlation of ingoing degree sequence and outgoing degree sequence for the simulated series with different parameters is consistent with the complexity and autocorrelation behavior in the corresponding definition of the time series. For the empirical analysis, VGDCCA method declares the similarity and dissimilarity between stock indices on time series irreversibility and captures the time irreversibility of traffic time series recorded by the detectors in different locations.

Our randomized versions of the Sharkovsky-type cycle coexistence theorems on tori and, in particular, on the circle are applied to random impulsive differential equations and inclusions. The obtained effective coexistence criteria for random subharmonics with various periods are formulated in terms of the Lefschetz numbers (in dimension one, in terms of degrees) of the impulsive maps and their iterates w.r.t. the (deterministic) state variables. Otherwise, the forcing properties of certain periods of the given random subharmonics are employed, provided there exists a random harmonic solution. In the single-valued case, the exhibition of deterministic chaos in the sense of Devaney is detected for random impulsive differential equations on the factor space $ℝ/\mathbb{Z}$. Several simple illustrative examples are supplied.

We study the minimum distance function of a complete intersection graded ideal in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we use the footprint function to give a sharp lower bound for the minimum distance function. Then we show some applications to coding theory.

Hydrogen bond is a key factor in the determination of structures and properties of room-temperature ionic liquids. Connections of these hydrogen bonds form a network. In this work, we analyzed the hydrogen bond network of 1-alkyl-3-methylimidazolium ionic liquids using network theory. A two-dimensional view of the hydrogen bond network has been generated, the connection pattern shown that the average length of line shape connection is 2.44 to 2.77 for six 1-alkyl-3-methylimidazolium ionic liquids, and the connection patterns are different for short and long alkyl side chain length. The degree of each ion was calculated and analyzed. The nodes with zero degree were adopted to detect the boundary of the clusters in the ionic liquids, which have no hydrogen bond connected with neighbor ions. This work indicates that the network analysis method is useful for understanding and predicting the structure and function of RTILs.

Let S be a semigroup. The degree of S is the smallest natural number r such that for each x ∈ S, x^n(x)+r = x^n(x), where n(x) ∈ ℕ. If such a number r does not exist, we say that the degree of S is infinite. For a group G, this coincides with the exponent of G. We prove that for a periodic ring R, the degree of R equals exp(U(R)), where U(R) denotes the unit group of R. Then we determine all degrees for any rings.

There are growing interests for studying collective behavior including the dynamics of markets, the emergence of social norms and conventions and collective phenomena in daily life such as traffic congestion. In our previous work [Iwanaga and Namatame, Collective behavior and diverse social network, International Journal of Advancements in Computing Technology4(22) (2012) 321–320], we showed that collective behavior in cooperative relationships is affected in the structure of the social network, the initial collective behavior and diversity of payoff parameter. In this paper, we focus on scale-free network and investigate the effect of number of interactions on collective behavior. And we found that choices of hub agents determine collective behavior.