Degree Sum Research Articles

QSPR analysis of some novel neighbourhood degree-based topological descriptors

Topological index is a numerical value associated with a chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. In this work, some new indices based on neighborhood degree sum of nodes are proposed. To make the computation of the novel indices convenient, an algorithm is designed. Quantitative structure property relationship (QSPR) study is a good statistical method for investigating drug activity or binding mode for different receptors. QSPR analysis of the newly introduced indices is studied here which reveals their predicting power. A comparative study of the novel indices with some well-known and mostly used indices in structure-property modelling and isomer discrimination is performed. Some mathematical properties of these indices are also discussed here.

On the Path Cover Number of Connected Quasi-Claw-Free Graphs

Detecting vertex disjoint paths is one of the central issues in designing and evaluating an interconnection network. It is naturally related to routing among nodes and fault tolerance of the network. A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths, and a path cover number of G denoted by p(G) = min{|P| : P is a path cover of G}. In this paper, we show that if the minimum degree sum of an independent set with k + 1 vertices in a connected quasi-claw-free graph G of order n is no less than n - k, then p(G) ≤ k - 1, where k ≥ 2. Examples illustrate that the degree sum condition in our result is sharp.

Multi-objective Dynamic Scheduling of Circuit Repair Based on Improved NSGA-II

To deal with the problem of tight wartime repair time, heavy repair task, and complicated constraints during wartime, the multi-objective dynamic scheduling study of circuit repair equipment maintenance task scheduling was carried out. The military problem of equipment maintenance task scheduling for circuit repair under complex constraints was proposed. Considering the uncertainty of the repair status of the equipment to be repaired, the constraints such as repair time window, non-traversal, and variety of repair capability were introduced on the basis of the repair capability and the repair time limit. A multi-objective dynamic scheduling model that takes the total number of repaired equipment, the sum of equipment importance degree, and the total time of secondary operations as the scheduling target.is established. An improved multi-objective genetic algorithm based on NSGA-II is designed to solve the model, and the rationality and effectiveness of the model and algorithm are verified through example simulation and analysis.

Minimum Degree Distance of Five Cyclic Graphs

Let G be a connected graph with n vertices. Then the class of connected graphs having n vertices is denoted by Gn. The subclass of connected graphs with 5 cycles are denoted by Gn5. The classification of graph G∈Gn5 depends on the number of edges and the sum of the degrees of the vertices of the graph. Any graph in Gn5 contains five linearly independent cycles having at least n+3 edges and the sum of degrees of vertices of 5-cyclic must be equal to twice of n+4. In this paper, minimum degree distance of class of five cyclic connected graph is investigated. To find minimum degree distance of a graph some transformations T have been defined. These transformation have been applied on the graph G∈Gn5 in such a way that the resultant graph belongs to Gn5 and also degree distance of T(G) is always must be less than G. For n=5, the five 5-cyclic graph has minimum degree distance 78 and the minimum degree distance of 5-cyclic graphs having six vertices is 124. In case of n greater than 6, a general formula for minimum degree distance is investigated. In this paper, we proved that the minimum degree distance of connected 5 cyclic graphs is 3n2+13n-62 by using transformations, for n≥7.

О новом предикторе, влияющем на ледообразование в Охотском море

Heat exchange between the three media – water, ice and air – must be taken into account when predicting the ice dynamics, drift and redistribution. It is known that the components of the heat balance vary quite strongly depending on the boundary of which media they are considered. Evaporation and turbulent heat exchange with the atmosphere are great in the areas of pure water, while evaporation from the surface of ice and snow is much less pronounced. To study the appearance of ice, it is necessary to consider only those environments between which intense heat exchange takes place; these environments are water and the atmosphere. This article studies the thermodynamic processes occurring over the seawater area by the statistical method of correlation analysis using the data on air temperature collected at hydrometeorological stations and those on the ice area from open sources. A new predictor is proposed, indicating a high correlation of 0.90–0.95 between the sum of degrees of daily temperature difference at the hydrometeorological stations of Okha and Oymyakon and the data on the area of the sea ice in the northwestern region of the Sea of Okhotsk on the last day of the month.

Molecular descriptors of neural networks with chemical significance

The probabilistic neural networks (PNNs) are now being analysed to fix a variety of challenges in the diverse fields of science and technology. In chemical graph theory, there are several tools, such as polynomials, functions, etc. that can be used to characterize different network properties. The neighborhood M-polynomial (NM) is one of those that yields neighborhood degree sum based topological indices in a manner that is less time consuming than the usual approach. In this work, the NM-polynomial of 3-layered and 4-layered probabilistic neural networks are derived. Further, some neighborhood degree sum based topological indices are computed from those polynomials. Applications of the present work are interpreted by investigating the chemical importance of the indices. Some structure property models are derived. The graphical representations of the results are also reported.

Generalized hypergraph coloring

A smooth hypergraph property $\mathcal{P}$ is a class of hypergraphs that is hereditary and non-trivial, i.e., closed under induced subhypergraphs and it contains a non-empty hypergraph but not all hypergraphs. In this paper we examine $\mathcal{P}$-colorings of hypergraphs with smooth hypergraph properties $\mathcal{P}$. A $\mathcal{P}$-coloring of a hypergraph $H$ with color set $C$ is a function $\varphi:V(H) \to C$ such that $H[\varphi^{-1}(c)]$ belongs to $\mathcal{P}$ for all $c \in C$. Let $L: V(H) \to 2^C$ be a so called list-assignment of the hypergraph $H$. Then, a $(\mathcal{P},L)$-coloring of $H$ is a $\mathcal{P}$-coloring $\varphi$ of $H$ such that $\varphi(v) \in L(v)$ for all $v \in V(H)$. The aim of this paper is a characterization of $(\mathcal{P},L)$-critical hypergraphs. Those are hypergraphs $H$ such $H-v$ is $(\mathcal{P},L)$-colorable for all $v \in V(H)$ but $H$ itself is not. Our main theorem is a Gallai-type result for critical hypergraphs, which implies a Brooks-type result for $(\mathcal{P},L)$-colorable hypergraphs. In the last section, we prove a Gallai bound for the degree sum of $(\mathcal{P},L)$-critical locally linear hypergraphs.

Spanning Trees with Few Peripheral Branch Vertices

Let $T$ be a tree, a vertex of degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. The set of leaves of $T$ is denoted by $L(T)$ and the set of branch vertices of $T$ is denoted by $B(T)$. For two distinct vertices $u$, $v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and $v$. Let $T$ be a tree with $B(T) \neq \emptyset$, for each vertex $x \in L(T)$, set $y_x \in B(T)$ such that $(V(P_T[x,y_x]) \setminus \{y_x\}) \cap B(T) = \emptyset$. We delete $V(P_T[x,y_x]) \setminus \{y_x\}$ from $T$ for all $x \in L(T)$. The resulting graph is a subtree of $T$ and is denoted by $\operatorname{R\_Stem}(T)$. It is called the reducible stem of $T$. A leaf of $\operatorname{R\_Stem}(T)$ is called a peripheral branch vertex of $T$. In this paper, we give some sharp sufficient conditions on the independence number and the degree sum for a graph $G$ to have a spanning tree with few peripheral branch vertices.

Degree sums and dominating cycles

A cycle C of a graph G is dominating if any vertex of V(G)∖V(C) has at least one neighbor on C and V(G)∖V(C) is an independent set. Let G be a k-connected graph of order n≥3 with k≥2. In this paper, we show that every longest cycle of G is dominating if the degree sums is more than (k+1)(n+1)∕3 for any k+1 pairwise nonadjacent vertices, and the lower bound is sharp, which generalizes the results due to Bondy (1980) for k=2 and Lu et al. (2005) for k=3.

The t-latency bounded strong target set selection problem in some kinds of special family of graphs

For a given simple graph $$G=(V,E)$$ , a latency bound t and a threshold function $$\theta (v)=\lceil \rho d(v)\rceil $$ , where $$\rho \in (0,1)$$ and d(v) denotes the degree of the vertex $$v(\in V)$$ , a subset $$S\subseteq V$$ is called a strong target set if for each vertex $$v\in S$$ , the number of its neighborhood in S not including itself is at least $$\theta (v)$$ , and all vertices in V can be activated by S through a process with t rounds. Initially, all vertices in S become activated. At the ith round of the process, each vertex is activated if the number of active vertices in its neighbor after $$i-1$$ rounds exceeds its threshold. The $$t$$ -Latency Bounded Strong Target Set Selection (t-LBSTSS) problem is to find such a strong target set S with the minimum cardinality in G. In general graphs, the t-LBSTSS problem is not only NP-hard, but also hard to be approximated. The aim of this paper is to find an optimal t-latency bounded strong target set for some special family of graphs. For a given simple graph G, a simple, tight but nontrivial inequality in terms of the number of edges in G is proposed to obtain the lower bound of the sum of degrees in a strong target set S to the t-LBSTSS problem. Moreover, a necessary and sufficient condition is presented for equality to hold. Finally, we give the exact formulas for the optimal solutions to the t-LBSTSS problem in two kinds of natural family of graphs, while it seems difficult to tell without the inequality given in this paper.

On reformulated Narumi-Katayama index

A graph is a mathematical model form by set of dots for vertices some of which are connected by lines named as edges. A topological index is a numeric value obtained from a graph mathematically which characterize its topology. The reformulated Narumi-Katayama index of a graph G is defined as the product of edge degrees of all the vertices of G which is introduced in 1984, to used the carbon skeleton of a saturated hydrocarbons. The degree of an edge is given by the sum of degrees of the end vertices of the edge minus 2. In this paper, we compute the reformulated Narumi-Katayama index for different graph operations.

Topological Properties of Para-Line Graph of Some Convex Polytopes Using Neighborhood M-Polynomial

The neighborhood M-polynomial is effective in recovering neighborhood degree sum based topological indices that predict different physicochemical properties and biological activities of molecular structures. Topological indices can transform the information found in molecular graphs and networks into numerical characteristics and thus make a major contribution to the study of structure-property and structure-activity relationships. In this work, the neighborhood M-polynomial of the para-line graph of some convex polytopes is obtained. From the neighborhood M-polynomial, some neighborhood degree-based topological indices are recovered. Applications of the work are described. In addition, a quantitative and graphical comparison is made.

On Degree Sum Conditions and Vertex-Disjoint Chorded Cycles

In this paper, we consider a general degree sum condition sufficient to imply the existence of k vertex-disjoint chorded cycles in a graph G. Let $$\sigma _t(G)$$ be the minimum degree sum of t independent vertices of G. We prove that if G is a graph of sufficiently large order and $$\sigma _t(G)\ge 3kt-t+1$$ with $$k\ge 1$$ , then G contains k vertex-disjoint chorded cycles. We also show that the degree sum condition on $$\sigma _t(G)$$ is sharp. To do this, we also investigate graphs without chorded cycles.

On degree sum conditions for directed path-factors with a specified number of paths

A directed path-factor of a digraph is a spanning subdigraph consisting of a union of vertex-disjoint directed paths in the digraph. In this paper, we give the following result: If D is a digraph of order n≥(2ℓ−1)k−1, and if dD+(u)+dD−(v)≥n−k for every two distinct vertices u and v with (u,v)∉A(D), then D has a directed path-factor with exactly k directed paths of order at least ℓ(≥2). To show this theorem, we discuss the correspondence between digraphs and bipartite graphs with perfect matchings, and also consider degree conditions for the existence of long directed paths in digraphs.

Does the enrichment of post-arable soil with organic matter stimulate forest ecosystem restoration—A view from the perspective of three decades after the afforestation of farmland

The restoration of forest ecosystems on former agricultural land faces numerous challenges resulting mainly from differences in soil properties between post-arable and continuously forested ground. We revisited a 34-year-old but very well-documented experiment with enrichment in organic matter of the arable sandy soil aiming to accelerate humus formation and soil biota regeneration in Scots pine plantation. The goal of our study was to determine whether the addition of a mixture of pine bark with sawdust or lupine cultivation would change the selected soil properties and, as a result, would shift soil microarthropods of post-agricultural plantations towards those occurring in pine monocultures on continuously forested land. The effect of tillage depth was also studied.The soil properties of the experimental plantation differed significantly from those of the forest soil with respect to the pH, nitrogen (N) content, C/N ratio, sum of exchangeable base cations and degree of soil saturation with base cations. Moreover, the diversity and taxonomic and functional structures of Collembola assemblages differed considerably from the assemblages of pine monoculture of the same age growing on continuously forested land. The expectation of the experiment was not met; however, a certain significant ameliorative effect on some soil properties was observed with some treatments. The total organic carbon content (TOC), C/N ratio and cation exchange capacity increased as a result of the addition of bark and sawdust. In plots where deep tillage was applied, the TOC and N contents were lower than those in plots with shallow soil cultivation. A significant increase in the number of hemiedaphic forms at the expense of euedaphic and atmobiotic forms was observed in plots with lupine, while a mixture of pine bark and sawdust caused an increase in the number of euedaphic forms.Soil pH was identified as a key parameter that determined the difference between forest and post-agricultural collembolan assemblages. The variability of the structure of post-agricultural assemblages depends on N and exchangeable Mg2+ contents and hydrolytic acidity. The enrichment of post-arable soil with organic matter and tillage had long-lasting effects on soil properties but did not stimulate restoration of forest collembolan assemblages on former farmland.

A novel entropy proposition for spherical fuzzy sets and its application in multiple attribute decision‐making

The inherent vagueness and uncertainty in reaching decisions are effectively coped with fuzzy logic and the concept of spherical fuzzy sets is one of the latest developments in this area. The hesitancy of decision-maker(s) about an attribute can be represented more extensively since the squared sum of membership, nonmembership, and hesitancy degrees should be between 0 and 1 while each degree should be defined in [0, 1]. In this study, we propose a novel entropy measure for spherical fuzzy sets and show its ability to provide the required properties. Then its usage in determining objective attribute weights is shown in an application of SF-WASPAS (Spherical Fuzzy extension of Weighted Aggregated Sum Product Assessment) on an illustrative problem. The robustness of the novel entropy's usage as an objective weighting tool is demonstrated by comparing the ranking solution of the proposed SF-WASPAS with the rankings obtained by SF-TOPSIS, SF-VIKOR, and SF-CODAS.

Spherical fuzzy extension of DEMATEL (SF‐DEMATEL)

In recent years, many versions of fuzzy sets are introduced in the literature. Spherical fuzzy sets (SFSs) concept is one of these latest developments. An SFS is a synthesis of Pythagorean fuzzy set (PFS) and neutrosophic set for providing a larger preference domain for experts and allowing them to express their hesitancy more comprehensively. The distinctive feature of SFSs is that the squared sum of membership, nonmembership, and hesitancy degrees is between 0 and 1 while each degree should be independently defined in [0,1]. The application of SFS on decision-making field may be about describing expert evaluations more informatively and explicitly. In this study, spherical fuzzy version of DEMATEL, one of the most cited multiple criteria decision-making approach, is introduced for considering the hesitancy degrees of experts when they evaluate the potential influences among the criteria. Another contribution made is the fuzzification of all the steps in DEMATEL until the last ones since the existing propositions in literature perform early defuzzification operations. The defuzzification is here applied as close to the end as possible for keeping the whole process fuzzy. The applicability of the proposition is shown in an illustrative example of a building contractor evaluation problem, and its robustness is shown by comparing the results with PFS-based and neutrosophic DEMATEL versions.

Color degree sum conditions for properly colored spanning trees in edge-colored graphs

For a vertex v of an edge-colored graph, the color degree of v is the number of colors appeared in edges incident with v. An edge-colored graph is called properly colored if no two adjacent edges have the same color. In this paper, we prove that if the minimum color degree sum of two adjacent vertices of an edge-colored connected graph G is at least |G|, then G has a properly colored spanning tree. This is a generalization of the result proved by Cheng, Kano and Wang. We also show the sharpness of this lower bound of the color degree sum.

On the maximum vertex degree in a conditional configuration graph

We consider configuration graphs under the condition that the sum of vertex degrees is bounded from above by n. The vertex degrees are independent identically distributed random variables. The distribution of the vertex degree ξ is unknown and has the only limiting condition that P{ξ=k} пропорционально d/( k g ln h k), where d>0, g>1, h≥0. We obtained the limit distributions of the maximum vertex degree as the number of graph vertices and n tends to infinity.

On the asymptotics of the degree structure of conditional internet graphs

We consider configuration graphs with N vertices, whose degrees are independent and identically distributed according to an unknown distribution law that depends on an arbitrary slowly varying function. The degree of each vertex is equal to the number of incident numbered semi-edges. The graph is constructed by joining all of the semi-edges pairwise equiprobably to form edges. Such models can be used to adequately describe various communication networks and Internet topologies. We study the subset of such random graphs under the condition that the sum of vertex degrees is known and it is equal to n. The paper finds the limit distributions of the maximum vertex degree and the number vertices with a given degree as N, n → ∞ in such a way that 0 < C 1 ≤ n/N ≤ C 2 < ∞.