Sombor index was recently defined in 2021 as a new vertex degree based topological index by Gutman and has received great attention of mathematicians and chemists. In this work, we determine how much the Sombor index is effected when we delete a vertex or an edge from a graph. Using the formulae obtained here successively, one can calculate the Sombor indices of large graphs by means of Sombor indices of smaller graphs which are obtained by deletion of vertices or edges.

A novel domination invariant defined by Boutrig and Chellali in the recent: total vertex-edge domination. In this paper we obtain an improved upper bound of total vertex edge-domination number of a tree. If is a connected tree with order , then with and we characterize the trees attaining this upper bound. Furthermore we provide a characterization of trees with

Constructing new graph from the graph's parameters and related notions in the way that, the study on the new graph and old graph in their parameters could be facilitated. As graph, new graph has some characteristics and results which are related to the structure of this graph. For this purpose, regular graph is considered so the internal relation and external relation on this new graph are studied. The kind of having same number of edges when this number is originated by common number of graphs like maximum degree, minimum degree, domination number, coloring number and clique number, is founded in the word of having regular graph

The Cohen-Macaulay vertex splittable ideals are characterized. As a consequence, we recover several Cohen-Macaulay classifications of families of monomial ideals known in the literature by new simpler combinatorial proofs.

New setting is introduced to study quasi-degree and quasi-co-degree arising from co-neighborhood. quasi-degree and quasi-co-degree is about a vertex which are applied into the setting of neutrosophic graphs. . The structure of set is studied and general results are obtained. Also, some classes of neutrosophic graphs namely path-neutrosophic graphs, cycle-neutrosophic graphs, complete-neutrosophic graphs and star-neutrosophic graphs, complete-bipartite-neutrosophic graphs and complete-multipartite-neutrosophic graphs are investigated in the terms of a vertex which is called either quasi-degree or quasi-co-degree. Neutrosophic number is reused in this way. It’s applied to use the type of neutrosophic number in the way that, three values of a vertex are used and they’ve same share to construct this number to compare with other vertices. Summation of three values of vertex makes one number and applying it to a comparison. This approach facilitates identifying vertices which form quasi-degree and quasi-co-degree. Quasi-degree is a value of a vertex which is maximum amid all values of vertices which are neighbors to a fixed vertex. Quasi-co-degree is a value of an edge which is maximum amid all values of edges which are neighbors to a fixed vertex but corresponded vertex is representative for this notion. Using different values which are related to a vertex inspire us to focus on edge and vertices which are corresponded to a fixed vertex. The notion of neighborhood is used to collect either vertices are titled neighbors or edges are incident to fixed vertex. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definitions are provided. Using fixed vertex has key role to have these notions in the form of vertex or edge. The value of an edge has eligibility to call quasi-co-degree but the value of a vertex has eligibility to call quasi-degree. Some results get more frameworks and perspective about these definitions. The way in that, two vertices have connection together, open the way to define neighborhood and co-neighborhood. The maximum values in neighborhood and co-neighborhood introduces quasi-degree and quasi-co-degree, respectively. New name is chosen from degree. Since amid all vertices with different degrees, one vertex is chosen. In other words, one vertex is fixed and its degree turns out quasi-degree where two degrees could be assigned to a vertex. Degree of edges and degree of vertices. The number of edges which are incident to the vertex and the number of vertices which are neighbors to the vertex. Degree and co-degree are the notions which are transformed to use in quasi-style. Two neutrosophic values introduce two neutrosophic vertices separately in each settings. These notions are applied into neutrosophic graphs as individuals but not family of them as drawbacks for these notions. Finding special neutrosophic graphs which are well-known, is an open way to purse this study. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this article.

The connected vertex cover (CVC) problem is a variant of the vertex cover problem, which has many important applications, such as wireless network design, routing and wavelength assignment problem, etc. A good algorithm for the problem can help us improve engineering efficiency, cost savings and resources in industrial applications. In this work, we present an efficient algorithm GRASP-CVC (Greedy Randomized Adaptive Search Procedure for Connected Vertex Cover) for CVC in general graphs. The algorithm has two main phases, i.e., construction phase and local search phase. To construct a high quality feasible initial solution, we design a greedy function and a restricted candidate list in the construction phase. The configuration checking strategy is adopted to decrease the cycling problem in the local search phase. The experimental results demonstrate that GRASP-CVC is competitive with the other competitive algorithm, which validate the effectivity and efficiency of our GRASP-CVC solver.

New setting is introduced to study failed zero-forcing number and failed zero-forcing neutrosophic-number. Leaf-like is a key term to have these notions. Forcing a vertex to change its color is a type of approach to force that vertex to be zero-like. Forcing a vertex which is only neighbor for zero-like vertex to be zero-like vertex but now reverse approach is on demand which is finding biggest set which doesn’t force. LetNTG : (V,E,σ,μ) be a neutrosophic graph. Then failed zero-forcing number Z(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximal cardinality of a set S of black vertices (whereas vertices in V (G) \ S are colored white) such that V (G) isn’t turned black after finitely many applications of “the color-change rule”: a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. Failed zero-forcing neutrosophic-number Zn(NTG) for a neutrosophic graphNTG : (V,E,σ,μ) is maximal neutrosophic cardinality of a set S of black vertices (whereas vertices in V (G) \ S are colored white) such that V (G) isn’t turned black afterfinitely many applications of “the color-change rule”: a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. Failed zero-forcing number and failed zero-forcing neutrosophic-number are about a set of vertices which are applied into the setting of neutrosophic graphs. The structure of set is studied and general results are obtained. Also, some classes of neutrosophic graphs namely path-neutrosophic graphs, cycle-neutrosophic graphs, complete-neutrosophic graphs, star-neutrosophic graphs, bipartite-neutrosophic graphs, and t-partite-neutrosophic graphs are investigated in the terms of maximal set which forms both of failed zero-forcing number and failed zero-forcing neutrosophic-number. Neutrosophic number is reused in this way. It’s applied to use the type of neutrosophic number in the way that, three values of a vertex are used and they’ve same share to construct this number to compare with other vertices. Summation of three values of vertex makes one number and applying it to a comparison. This approach facilitates identifying vertices which form failed zero-forcing number and failed zero-forcing neutrosophic-number. In path-neutrosophic graphs, the set of vertices such that every given two vertices in the set, have distance at least two, forms maximal set but with slightly differences, in cycle-neutrosophic graphs, the set of vertices such that every given two vertices in the set, have distance at least two, forms maximal set. Other classes have same approaches. In complete-neutrosophic graphs, a set of vertices excluding two vertices leads us to failed zero-forcing number and failed zero-forcing neutrosophic-number. In star-neutrosophic graphs, a set of vertices excluding only two vertices and containing center, makes maximal set. In complete-bipartite-neutrosophic graphs, a set of vertices excluding two vertices from same parts makes intended set but with slightly differences, in complete-t-partite-neutrosophic graphs, a set of vertices excluding two vertices from same parts makes intended set. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. Using basic set not to extend this set to set of all vertices has key role to have these notions in the form of failed zero-forcing number and failed zero-forcing neutrosophic-number. The cardinality of a set has eligibility to form failed zero-forcing number but the neutrosophic cardinality of a set has eligibility to call failed zero-forcing neutrosophic-number. Some results get more frameworks and perspective about these definitions. The way in that, two vertices don’t have unique connection together, opens the way to do some approaches. A vertex could affect on other vertex but there’s no usage of edges. These notions are applied into neutrosophic graphs as individuals but not family of them as drawbacks for these notions. Finding special neutrosophic graphs which are well-known, is an open way to pursue this study. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this article.

In this article, some kinds of triple belongs to metric dimensions are defined. Some classes of graphs in the matter of these kinds, are studied and the relation amid these kinds are considered. The kind of having equivalency amid these notions and some classes of graphs, is obtained. The kind of locating some vertices by some vertices when the number of locating vertices is increased, has the key role to analyze the classes of graphs, general graphs, and graph's parameters.

Mechanical equipment is composed of several parts, the interaction between the parts exists throughout whole life cycle leads to the widespread phenomenon of fault coupling and the diagnosis of independent faults cannot meet the requirements of the health management of mechanical equipment under actual working conditions. In this paper, the topological structure of graph neural network is used to describe the correlation of coupling faults and multiple fault types are regarded as network nodes respectively. The fault features are fused and extracted by graph convolution to achieve the classification of coupling faults. In this paper, dynamic vertexes are defined in data topology and the parameters of DIGNN classification model are optimized by changing the fault type of dynamic vertexes during training stage. The data is loaded into dynamic vertexes for classification and analysis during testing stage. The one-dimensional vibration data is converted into two-dimensional time-frequency domain data by wavelet transform making the input features of the model interpretable to reduce the uncertainty of model training. The data topology is then fed into DIGNN for fault classification, and GCN's node aggregation gives interpretability to each layer of network data processing. The method proposed in this paper can realize accurate diagnosis of independent faults on the data set, and can effectively judge the coupling mode of coupling faults, which is suitable for coupling fault analysis of mechanical equipment.

We introduce a new variation on the domination theme which we call vertex domination as reducing waste of time in transportation planning and optimization of transport routes. We determine the vertex domination number $\gamma_v$ for several classes of fuzzy graphs. The bounds is obtained for it. In fuzzy graphs, monotone decreasing property and monotone increasing property are introduced. We prove both of the vizing's conjecture and the Grarier-Khelladi's conjecture are monotone decreasing fuzzy graph property for vertex domination. We obtain Nordhaus-Gaddum (NG) type results for these parameters. The relationship between several classes of operations on fuzzy graphs with the vertex domination number of them is studied. Finally, we discuss about vertex dominating set of a fuzzy tree by using the bridges and $\alpha$-strong edges equivalence.

In this paper, we examine the interplay between the structural and spectral properties of the unit graph G(Zn) for n=p1r1p2r2...pkrk, where p1,p2,...,pk are distinct primes and k,r1,r2,...,rk are positive integers such that at least one of the ri must be greater than 1. We first analyze the structure of the unit graph of Zn as a generalized join graph under these conditions. We then determine the Laplacian spectrum of G(Zn) and deduce that it is integral for all n. Consequently, we obtain Laplacian spectral radius and algebraic connectivity of G(Zn). We also prove that the vertex connectivity of G(Zpq) is (p−2)q, where 2≠p<q. We deduce the vertex connectivity of G(Zn) when n=prqs, where 2≠p<q are primes and r,s are positive integers. Finally, we present conjectures about the vertex connectivity of G(Zn) when n=p1p2...pk and n=p1r1p2r2...pkrk, where pi are distinct primes, ri are positive integers, and 1≤i≤k.

Topological indices are invariant numerical fields of a graph that give facts about the structure of graphs and are found to be very helpful in predicting the physical properties of aluminophosphates. The characteristics of the aluminophosphates are similar to those of zeolites. Two examples of current applications are natural gas dehydration and humidity sensor. New frameworks are being synthesised by researchers in chemistry and materials science. There are many layers and holes in these substances. In this study, Vertex version of distance-based topological indices, the entropy of topological indices and their numerical analysis are explained.

The generalization of Farey graphs and extended Farey graphs are all originated from Farey graph and scale-free and small-world simultaneously. We propose a labeling of the vertices for it that allows determining all the shortest paths routing between any two vertices only based on their labels. The maximum number of shortest paths between any two vertices is huge as the product of two Fibonacci numbers, however, the label-based routing algorithm runs in linear time O(n). The existence of an efficient routing protocol for Farey-type models should help the understanding of several physical dynamic processes on it.

Physical symptoms are defined as symptoms for which adequate examination does not reveal a sufficient underlying root cause, e.g., pain and fatigue. The extant literature of the neurobiological underpinnings of physical symptoms has been largely inconsistent and primarily consists of (clinical) case-control studies with relatively small samples sizes. Therefore, we studied the association of brain morphology with physical symptoms in pre-adolescents from two independent and population-based cohorts. This study included 2,683 individuals from the Generation R Study (51% girls, 10.1 ± 0.6 years old) and 10,567 pre-adolescents from the ABCD Study (48% girls, 9.9 ± 0.6 years old). High- resolution structural magnetic resonance imaging (MRI) was collected using 3-Tesla MRI systems. Physical symptoms were evaluated using the somatic complaints syndrome scale from the parent-reported school-age version of the Child Behavior Checklist. Linear regression models were fitted for global brain metrics (i.e., cortical and subcortical grey matter volume and total white matter volume) as well as surface-based vertex-wise measures (surface area and cortical thickness). Analyses were initially conducted separately in each cohort and later meta-analysed. No associations were observed in either cohort separately. In the combined vertex-wise meta-analysis of both cohorts; the right hemisphere surface area, most notably the rostral middle frontal gyrus, superior frontal gyrus and anterior cingulate cortex, were related to physical symptoms after correcting for multiple comparisons (cluster area = 1,882 mm2). The present study, which is the most representative and well-powered to date, suggests that surface area, but not other measures of brain morphology, are modestly related to physical symptoms in pre- adolescents. While these effects are subtle, future longitudinal research is warranted to elucidate whether such associations indicate a cause or a consequence of the physical symptoms.

Due to the fact that negative energies have no existence in physical reality, the advanced mechanics of purely positive energies should describe gravitational interactions and collisions in monistic terms of extended kinetic energies and their local stresses. Such non-Newtonian mechanics of continuous inertial densities reinforces the Cartesian concept of matter-extension in the metric formalism of Einstein-Grossmann with a supplemental (dark, aether) fraction of bi-vertex mass-energy distributions. Local accelerations or decelerations of mono-vertex material densities in a multi-vertex distribution of complete kinetic energy arise under its constant integral due to nonlocal organization of continuous densities. Such integral conservation of the distributed mass-energy occurs instantaneously throughout the whole continuum of correlated densities and metric stresses despite the time-varying contributions of complementary mono-vertex and bi-vertex fractions. Under the nonlocal organization of purely kinetic (positive) mass-energy, geodesic self-heating and self-cooling of the pulsating space-matter conserve the integral energy in the two-fraction virial theorem for the averaged motion of visible mono-vertexes in the presence of invisible bi-vertex (interference, dark) mass-energy. Metric stresses of such material space are subordinate to nonlocal self-government of continuously distributed kinetic energy, including the relativistic rest-energy of General Relativity. These mutually consistent or correlated stresses in inertial space-time-energy create timelessly coordinated self-accelerations, observed for dense material volumes as distant gravitational pulls. In order to falsify/verify the nonlocal self-organization of adaptive kinetic energy, the monistic mechanics of self-consistent inertial densities and metric stresses can suggest moderate field changes in the temporal redshift, cycles of geodetic falls and takeoffs in pulsating kinetic organizations, and the calculated acceleration of the expanding Metagalaxy in its current phase of geodesic self-cooling.

The minimum vertex cover (MVC) problem is a canonical NP-hard combinatorial optimization problem, aiming to find a smallest set of vertices such that every edge has at least one endpoint in the set, which has extensive applications in cyber security, scheduling, and monitoring link failures in wireless sensor networks (WSNs). Numerous local search algorithms have been proposed to obtain a “good” vertex cover. However, due to the NP-hard nature, it is challenging to efficiently solve the MVC problem, especially on large graphs. In this paper, we propose an efficient local search algorithm for MVC called TIVC, which is based on two main ideas: A 3-improvements (TI) framework with tiny perturbation, and an edge selection strategy. We conducted experiments on real-world large instances of a massive graph benchmark. Compared with two state-of-the-art MVC algorithms, TIVC shows superior performance in accuracy and possesses a remarkable ability to identify significantly smaller vertex covers on many graphs.

For the first time, We do fuzzification the concept of domination in crisp graph on a generalization of fuzzy graph by using membership values of vertices, α-strong edges and edges. In this paper, we introduce the first variation on the domination theme which we call vertex domination. We determine the vertex domination number γ_v for several classes of t-norm fuzzy graphs which include complete t-norm fuzzy graph, complete bipartite t-norm fuzzy graph, star t-norm fuzzy graph and empty t-norm fuzzy graph. The relationship between effective edges and α-strong edges is obtained. Finally, we discuss about vertex dominating set of a fuzzy tree with respect to a t-norm ⨂ by using the bridges and α-strong edges equivalence.

We introduce a new class of Vertex Operator Algebras Y+ and their duals, which generalize the standard W-algebras WN of type sl(N). These algebras can be defined in terms of junctions of boundary conditions and interfaces in the GL-twisted N = 4 Super Yang Mills gauge theory. The aim of these technical calculations is to find the relation of these ortho-symplectic Y-algebras to truncations of even W\infinity.

The real-time Railway Traffic Management Problem (rtRTMP) is discussed in the field of railway transportation. In this paper, rtRTMP is considered for planning shunting routes based on the use of routing methods on a spatial-temporal graph. An approach and algorithm of Spatial-Temporal Graph formation for rail yard transportation network is proposed. The idea is to construct an undirected loop-free Double Vertex Graph for determining optimal shunting routes consisting of several half-runs for several trains shunting under draw-out tracks capacity constraints. Experimental results on the rail yard simulation model are presented.

In this paper, we demonstrate a useful interaction between the theory of clique partitions, edge clique covers of a graph, and the spectra of graphs. Using a clique partition and an edge clique cover of a graph we introduce the notion of a vertex-clique incidence matrix for a graph and produce new lower bounds for the negative eigenvalues and negative inertia of a graph. Moreover, utilizing these vertex-clique incidence matrices, we generalize several notions such as the signless Laplacian matrix, and develop bounds on the incidence energy and the signless Laplacian energy of the graph. %The tight upper bounds for the energies of a graph and its line graph are given. More generally, we also consider the set $S(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent. An important parameter in this setting is $q(G)$, and is defined to be the minimum number of distinct eigenvalues over all matrices in $S(G)$. For a given graph $G$ the concept of a vertex-clique incidence matrix associated with an edge clique cover is applied to establish several classes of graphs with $q(G)=2$.

The independent set, IS, on a graph $G=(V,E)$ is $V^{*}\subseteq V$ such that no two vertices in $V^{*}$ have an edge between them. The MIS problem on G seeks to identify an IS with maximum cardinality, i.e. MIS. $V^{*}\subseteq V$ is a vertex cover, i.e. VC of $G=(V,E)$ if every $e\in E$ is incident upon at least one vertex in $V^{*}$ . $V^{*}\subseteq V$ is dominating set, DS, of $G=(V,E)$ if $\forall v\in V$ either $v\in V^{*}$ or $\exists u\in V^{*}$ and $(u,v)\in E$ . The MVC problem on G seeks to identify a vertex cover with minimum cardinality, i.e. MVC. Likewise, MCV seeks a connected vertex cover, i.e. VC which forms one component in G, with minimum cardinality, i.e. MCV. A connected DS, CDS, is a DS that forms a connected component in G. The problems MDS and MCD seek to identify a DS and a connected DS i.e. CDS respectively with minimum cardinalities. MIS, MVC, MDS, MCV and MCD on a general graph are known to be NP-complete. Polynomial time algorithms are known for bipartite graphs, chordal graphs, cycle graphs, comparability graphs, claw-free graphs, interval graphs and circular arc graphs for some of these problems. We introduce a novel graph class, layered graph, where each layer refers to a subgraph containing at most some k vertices. Inter layer edges are restricted to the vertices in adjacent layers. We show that if $k=\Theta (log\mid V\mid )$ then MIS, MVC and MDS can be computed in polynomial time and if $k=O\left({(log\mid V\mid )}^{\alpha }\right)$ , where $\alpha <1$ , then MCV and MCD can be computed in polynomial time. If $k=\Theta \left({(log\mid V\mid )}^{1+ϵ}\right)$ , for $ϵ>0$ , then MIS, MVC and MDS require quasi-polynomial time. If $k=\Theta (log\mid V\mid )$ then MCV, MCD require quasi-polynomial time. Layered graphs do have constraints such as bipartiteness, planarity and acyclicity.