Sum Of Vertices Research Articles

Sombor topological indices for different nanostructures

Euclidean geometry is utilized to establish the Sombor graph parameter and its invariants. It is sum of all adjacent vertices in graph theory dϒ2+dΓ2 where dϒ is the degree of the vertex ϒ. Geometrical interpretation is used to describe the new Sombor indices types. We examined, recently developed Sombor indices for various nanotube Y-junctions in this article. In specifically, the first area-based Sombor index was introduced by Euclidean geometry. Angular orientation concept to construct the second, fourth, and sixth Sombor graph parameters, while third and fifth Sombor graph parameters are constructed by perimeter.

Пошук максимальних незалежних множин вершин графа для вдосконалення програмних проєктів

The need is fixed to software project enhancing with seamless integration of technological-descriptive and normative project manage- ment approaches by means of classical Graph Discrete Optimization Problems tailoring for software project management tasks, poorly equipped with best practices within technological approach. Class of software project management tasks is proposed to demonstrate the benefits of such integration. Two Boolean linear programming problems are investigated for searching some maximum size indepen- dent set (Section 1) and an algorithm for searching all possible maximum size independent sets (Section 2). Section 3 presents Problem Statement for searching a given number of non-intersecting independent sets with maximum sum of vertices’ numbers within independent sets. Based on it, Vizing-Plesnevich algorithm is described for coloring the graph vertices with the minimum number of colors. To solve Boolean problems, both specialized mathematical programming language AMPL and corresponding solver program named gu- robi are used. For basic algorithms developed, reference AMPL code versions are given as well as their running results. Illustrative examples of software project enhancing with the algorithms elaborated are considered in Section 4, namely: 25 specialists being conflicted during their previous projects partitioning into coherent conflict-free sub-teams for software projects portfolio; schedule optimization for autonomous testing of reusable components within a critical software system; cores composing for independent teams in a critical software project.

A REMARK ON THE EDGE IRREGULARITY STRENGTH OF CORONA PRODUCT OF TWO PATHS

With respect to a simple graph G, a vertex labeling ϕ: V(G) > {1,2,...,k) is known as k-labeling. The weight corresponding to an edge xy in G, expressed as wϕ (xy), represents the labels sum of end vertices x and y, given by wϕ (xy) = ϕ(x) + ϕ(y) A vertex k-labeling is expressed as an edge irregular k-labeling with respect to graph G provided that for every two distinct edges e and f, there exists wϕ(e) ≠ wϕ(f) Here, the minimum k where the graph G possesses an edge irregular k-labeling is known as the edge irregularity strength with respect to G, expressed as (G). Here, we examine the edge irregularity strength’s exact value of corona product with respect to two paths Pn and Pm , in which n ≥ 2 and m = 3, 4, 5.

Simplicial Dollar Game

The dollar game is a chip-firing game introduced by Baker as a context in which to formulate and prove the Riemann-Roch theorem for graphs. A divisor on a graph is a formal integer sum of vertices. Each determines a dollar game, the goal of which is to transform the given divisor into one that is effective (nonnegative) using chip-firing moves. We use Duval, Klivans, and Martin's theory of chip-firing on simplicial complexes to generalize the dollar game and results related to the Riemann-Roch theorem for graphs to higher dimensions. In particular, we extend the notion of the degree of a divisor on a graph to a (multi)degree of a chain on a simplicial complex and use it to establish two main results. The first of these generalizes the fact that if a divisor on a graph has large enough degree (at least as large as the genus of the graph), it is winnable; and the second generalizes the fact that trees (graphs of genus $0$) are exactly the graphs on which every divisor of degree $0$, interpreted as an instance of the dollar game, is winnable.

2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices

2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices

On The Local Edge Antimagic Coloring of Corona Product of Path and Cycle

Let be a nontrivial and connected graph of vertex set and edge set . A bijection is called a local edge antimagic labeling if for any two adjacent edges and , where for . Thus, the local edge antimagic labeling induces a proper edge coloring of G if each edge e assigned the color . The color of each an edge e = uv is assigned bywhich is defined by the sum of label both and vertices and . The local edge antimagic chromatic number, denoted by is the minimum number of colors taken over all colorings induced by local edge antimagic labeling of . In our paper, we present the local edge antimagic coloring of corona product of path and cycle, namely path corona cycle, cycle corona path, path corona path, cycle corona cycle.Keywords: Local antimagic; edge coloring; corona product; path; cycle.

The super local antimagic - decomposition coloring of graph

Let G be a nontrivial, finite, connected graph with vertex set V and edge set E. A bijection is called a local antimagic -decomposition labeling for any two adjacent subgraph and , , where . The local antimagic -decomposition labeling that induces a proper decomposition coloring of G where the decomposition is assigned by the color is called local antimagic -decomposition coloring. We call labeling of local antimagic -decomposition coloring is super if we add vertices label start from 1 until the sum of vertices, edges label start from the sum of vertices plus 1 until the sum of vertices and edges. The minimum number of colors in super local antimagic -decomposition coloring is super local antimagic -decomposition coloring chromatic number and denoted by γlatH(G). In this paper, we used some special graph such as wheel graph (Wn), sun graph Sun and star graph Sn

Local super antimagic total face coloring of planar graphs

We using graph G = (V(G), E(G), F(G)) be a nontrivial, finite, connected graph, and a g bijective function mapping total labeling of graph to natural number start form 1 until the sum of vertices, edge, and faces. The sum of vertices, edges, and faces labels in a face f is called the weight of the face f ∈ F(G). If any adjacent two faces f1 and f2 have different weights w(f1) ≠ w(f2) for f1, f2 ∈ F(G), then g is called a labeling of local antimagic total face. We call labeling of local antimagic total face is super if we add vertices label start from 1 until the sum of vertices, edges label start from the sum of vertices plus 1 until the sum of vertices and edges, and faces label start form the sum of vertices and edges plus one untul the sum of vertices, edges, and faces. The local super antimagic total face labeling that induces a proper faces coloring of G where the faces f is assigned by the color w(f) is called local super antimagic total face coloring. The minimum number of colors in local super antimagic total face coloring is local antimagic total face chromatic number and denoted by γlatf(G). In this paper, we used some planar graph such as wheel graph (Wn), jahangir graph (J(2, n)), ladder graph (Ln), and circular ladder graph (CLn). Our results attained the lower bound of local super antimagic total face chromatic number.

Generalized vertex cover using chemical reaction optimization

The generalized vertex cover problem (GVC) is a new variant of classic vertex cover problem which considers both vertex and weight of the edge into the objective function. The GVC is a renowned NP-hard optimization problem that finds the vertex subset where the sum of vertices and edge weight are minimized. In the mathematical field of electrical, networking and telecommunication GVC is used to solve the vertex cover problem. Finding the minimum vertex cover using GVC has a great impact on graph theory. Some exact algorithms were proposed to solve this problem, but they failed to solve it for real-world instances. Some approximation and metaheuristic algorithms also were proposed to solve this problem. Chemical Reaction Optimization (CRO) is an established population-based metaheuristic for optimization and comparing with other existing optimization algorithms it gives better results in most of the cases. The CRO algorithm helps to explore the search space locally and globally over the large population area. In this paper, we are proposing an algorithm by redesigning the basic four operators of CRO to solve GVC problem and an additional operator named repair function is used to generate optimal or near-optimal solutions. We named the proposed algorithm as GVC_CRO. Our proposed GVC_CRO algorithm is compared with the hybrid metaheuristic algorithm (MAGVCP), the local search with tabu strategy and perturbation mechanism (LSTP) and Genetic Algorithm (GA), which are state of the arts. The experimental results show that our proposed method gives better results than other existing algorithms to solve the GVC problem with less execution time in maximum cases. Statistical test has been performed to demonstrate the superiority of the proposed algorithm over the compared algorithm.

On the Total Edge Irregularity Strength of Generalized Butterfly Graph

Let G(V, E) be a connected, simple, and undirected graph with vertex set V and edge set E. A total k-labeling is a map that carries vertices and edges of a graph G into a set of positive integer labels {1, 2, …, k}. An edge irregular total k-labeling λ: V(G) ∪ E(G) → {1, 2, …, k} of a graph G is a total k-labeling such that the weights calculated for all edges are distinct. The weight of an edge uv in G, denoted by wt(uv), is defined as the sum of the label of u, the label of v, and the label of uv. The total edge irregularity strength of G, denoted by tes(G), is the minimum value of the largest label k over all such edge irregular total k-labelings. A generalized butterfly graph, BFn, obtained by inserting vertices to every wing with assumption that sum of inserting vertices to every wing are same then it has 2n + 1 vertices and 4n − 2 edges. In this paper, we investigate the total edge irregularity strength of generalized butterfly graph, BFn, for n > 2. The result is .

A robust and cooperative parallel tabu search algorithm for the maximum vertex weight clique problem

Abstract The maximum vertex weight clique problem (MVWCP) is a challenging NP-Hard combinatorial optimization problem that searches for a clique with maximum total sum of vertices’ weights. In this study, we propose a robust and cooperative parallel tabu search algorithm (PTC) for the MVWCP. Our proposed algorithm uses a dedicated tabu search algorithm with a multistart strategy for the diversification of search space on a parallel computation environment. An effective seeding mechanism is developed with respect to the rank of the processors to choose diversified starting points for a better exploration of the search space. Classical add, swap and drop operators of tabu search are improved with parallel computation and a combined neighborhood approach. The PTC algorithm is evaluated on a set of 120 problem instances from DIMACS-W and BHOSLIB-W benchmarks. Computational results show that the PTC algorithm competes with state-of-the-art heuristic algorithms by reporting average best (optimal) result hit ratios up to 99.0%.

Hyper Zagreb Index of Bridge and Chain Graphs

Let G be a simple connected molecular graph with vertex set $V(G)$ and edge set $E(G)$. One important modification of classical Zagreb index, called hyper Zagreb index $HM(G)$ is defined as the sum of squares of the degree sum of the adjacent vertices, that is, sum of the terms $[{d_G}(u)+{d_G}(v)]^2$ over all the edges of $G$, where ${d_G}(u)$ denote the degree of the vertex $u$ of $G$. In this paper, the hyper Zagreb index of certain bridge and chain graphs are computed and hence using the derived results we compute the hyper Zagreb index of several classes of chemical graphs and nanostructures.

Progress towards the total domination game [formula omitted]-conjecture

In this paper, we continue the study of the total domination game in graphs introduced in Henning et al. (2015), where the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a total dominating set S of G in which every vertex is totally dominated by a vertex in S. Dominator wishes to minimize the number of vertices chosen, while Staller wishes to maximize it. The game total domination number, γtg(G), of G is the number of vertices chosen when Dominator starts the game and both players play optimally. Henning et al. (in press) posted the 34-Game Total Domination Conjecture that states that if G is a graph on n vertices in which every component contains at least three vertices, then γtg(G)≤34n. In this paper, we prove this conjecture over the class of graphs G that satisfy both the condition that the degree sum of adjacent vertices in G is at least 4 and the condition that no two vertices of degree 1 are at distance 4 apart in G. In particular, we prove that by adopting a greedy strategy, Dominator can complete the total domination game played in a graph with minimum degree at least 2 in at most 3n/4 moves.

A State Polytope Decomposition Formula

AbstractWe give a decomposition formula for computing the state polytope of a reducible variety in terms of the state polytopes of its components: if a polarized projective variety X is a chain of subvarieties Xi satisfying some further conditions, then the state polytope of X is the Minkowski sum of the state polytopes of Xi translated by a vector τ, which can be readily computed from the ideal of Xi. The decomposition is in the strongest sense in that the vertices of the state polytope of X are precisely the sum of vertices of the state polytopes of Xi translated by τ. We also give a similar decomposition formula for the Hilbert–Mumford index of the Hilbert points of X. We give a few examples of the state polytope and the Hilbert–Mumford index computation of reducible curves, which are interesting in the context of the log minimal model program for the moduli space of stable curves.

Cutting-plane-based algorithms for two branch vertices related spanning tree problems

A branch vertex is a vertex with degree larger than or equal to three. This paper addresses two spanning tree problems in an undirected, simple graph. The first one is to find a spanning tree that minimizes the number of branch vertices (MBV), and the second one is to find a spanning tree that minimizes the degree sum of branch vertices (MDS). These two problems arise in the design of wavelength-division networks (WDN), when the cost of equipments for enabling multicast communication is to be minimized. After investigating the relations of MBV and MDS with the problem of minimizing the number of leaves in a spanning tree, new formulations based on ILP are proposed for MBV and MDS, along with two cutting plane algorithms for addressing them. A repair function is also introduced for deriving feasible solutions from the candidate trees returned at each iteration of the cutting plane algorithm, as well as a Tabu search procedure for further quality improvement. The resulting hybrid approach is shown to outperform pure ILP formulations and heuristic approaches published earlier.

Light subgraphs of graphs embedded in the plane—A survey

It is well known that every planar graph contains a vertex of degree at most 5. A theorem of Kotzig states that every 3-connected planar graph contains an edge whose endvertices have degree-sum at most 13. Fabrici and Jendrol’ proved that every 3-connected planar graph G that contains a k-vertex path contains also a k-vertex path P such that every vertex of P has degree at most 5k. A result by Enomoto and Ota says that every 3-connected planar graph G of order at least k contains a connected subgraph H of order k such that the degree sum of vertices of H in G is at most 8k−1. Motivated by these results, a concept of light graphs has been introduced. A graph H is said to be light in a family G of graphs if at least one member of G contains a copy of H and there is an integer w(H,G) such that each member G of G with a copy of H also has a copy K of H such that ∑v∈V(K)degG(v)≤w(H,G).In this paper we present a survey of results on light graphs in different families of plane graphs and multigraphs. A similar survey dealing with the family of all graphs embedded in surfaces other than the sphere was prepared as well.

New heuristics for two bounded-degree spanning tree problems

A vertex v of a connected graph G=(V,E) is called a branch vertex if its degree is greater than two. Pertaining to branch vertices, this paper studies two optimization problems having roots in the domain of optical networks. The first one, referred to as MBV, seeks a spanning tree T of G with the minimum number of branch vertices, whereas the second problem, referred to as MDS, seeks a spanning tree T of G with the minimum degree sum of branch vertices. Both MBV and MDS are NP-Hard. Two heuristics approaches are presented for each problem. The first approach is a problem specific heuristic, whereas the latter one is a hybrid ant-colony optimization algorithm. Computational results show the effectiveness of our proposed approaches.

Degree Bounded Spanning Trees

In this paper, we give a sufficient condition for a graph to have a degree bounded spanning tree. Let n ≥ 1, k ≥ 3, c ≥ 0 and G be an n-connected graph. Suppose that for every independent set $${S \subseteq V(G)}$$of cardinality n(k−1) + c + 2, there exists a vertex set $${X \subseteq S}$$of cardinality k such that the degree sum of vertices in X is at least |V(G)| − c −1. Then G has a spanning tree T with maximum degree at most $${k+\lceil c/n\rceil}$$and $${\sum_{v\in V(T)}\max\{d_T(v)-k,0\}\leq c}$$.

Approximating the Maximum Internal Spanning Tree problem

Given an undirected connected graph G we consider the problem of finding a spanning tree of G which has a maximum number of internal (non-leaf) vertices among all spanning trees of G . This problem, called Maximum Internal Spanning Tree problem, is clearly NP-hard since it is a generalization of the Hamiltonian Path problem. From the optimization point of view the Maximum Internal Spanning Tree problem is equivalent to the Minimum Leaf Spanning Tree problem. However, the two problems have different approximability properties. Lu and Ravi proved that the latter has no constant factor approximation–unless P = NP–, while Salamon and Wiener gave a linear-time 2-approximation algorithm for the Maximum Internal Spanning Tree problem. In this paper, we improve this approximation ratio by giving an O ( | V ( G ) | 4 ) -time 7 / 4 -approximation algorithm for graphs without pendant vertices. Our approach is based on the successive execution of local improvement steps. We use a linear programming formulation and a primal–dual technique to prove the approximation ratio. We also investigate the vertex-weighted case, that is to find a spanning tree of a vertex-weighted graph G in which the weight sum of internal vertices is maximal among all spanning trees of G . For this problem we present a ( 2 Δ ( G ) − 3 ) -approximation algorithm, where Δ ( G ) is the maximum vertex-degree of G . A slight modification of this algorithm ensures a 2-approximation whenever the input graph is claw-free. Both algorithms run in O ( | V ( G ) | 4 ) time for graphs with no pendant vertices.

Bounded-degree spanning tree problems: models and new algorithms

Given a connected graph G, a vertex v of G is said to be a branch vertex if its degree is greater than 2. We consider two problems arising in the context of optical networks: Finding a spanning tree of G with the minimum number of branch vertices and Finding a spanning tree of G such that the degree sum of the branch vertices is minimized. For these NP-hard problems, heuristics, that give good quality solutions, do not exist in the literature. In this paper we analyze the relation between the problems, provide a single commodity flow formulation to solve the problems by means of a solver and develop different heuristic strategies to compute feasible solutions that are compared with the exact ones. Our extensive computational results show the algorithms to be very fast and effective.