Study of θϕ Networks via Zagreb Connection Indices

Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network θϕ formed by ϕ time repetition of the process of joining θ copies of a selected graph Ω in such a way that corresponding vertices of Ω in all the copies are joined with each other by a new edge. The symmetry of θϕ is ensured by the involvement of complete graph Kθ in the construction process. The free hand to choose an initial graph Ω and formation of chemical graphs using θϕΩ enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of θϕΩ. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of θϕΩ to enable its emerging use.

Feedback and the emergence of structures in information systems.

The aim of the work is to find conditions under which a complex multifunctional system acquires a certain struc- ture and comes to a balanced state. The initial system, consisting of mutually influencing objects or processes, is modeled by a sign graph. Analysis of the degree of influence takes into account both the direct impact between neighboring peaks, and indirect, mediated through other objects. Examples are given and conditions are indicated under which, under the ac- tion of the mechanism of introducing feedbacks, an unbalanced initial graph of interactions comes to a stable two-class structure. Cases are shown when disturbances in a balanced graph, leading it out of this state, after the application of the feedback mechanism were leveled, and the system was stabilized. Both fully connected graphs and digraphs without cycles are investigated.

Valued Number And Set

In this essay, the new notion concerning longest path is introduced. Longest path has a close relation with the notion of diameter in graph. The classes of graph are studied in the terms of having the vertex with longest path. Valued number is the number of edges belong to the longest path in the matter of vertex. For every vertex, there’s a valued number and new notion of valued set is the generalization of valued number for the vertex when all vertices of the graphs are corresponded to a vertex which has the greater valued number. For any positive integer, there’s one graph in that, there’s vertex which its valued number is that. By deleting the vertices which don’t belong to valued set, new notion of new graph is up. It’s called valued graph. The comparison amid valued graph and initial graph is up, too.

NP-completeness of the Minimum Spanning Tree Problem of a Multiple Graph of Multiplicity k ≥ 3

In this paper, we study undirected multiple graphs of any natural multiplicity k > 1. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of k linked edges, which connect 2 or (k + 1) vertices correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of k linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. A multiple tree is a connected multiple graph with no cycles. Unlike ordinary trees, the number of edges in a multiple tree is not fixed. The problem of finding the spanning tree can be set for a multiple graph. Complete spanning trees form a special class of spanning trees of a multiple graph. Their peculiarity is that a multiple path joining any two selected vertices exists in the tree if and only if such a path exists in the initial graph. If the multiple graph is weighted, the minimum spanning tree problem and the minimum complete spanning tree problem can be set. Also we can formulate the problems of recognition of the spanning tree and complete spanning tree of the limited weight. The main result of this article is the proof of NPcompleteness of such recognition problems for arbitrary multiple graphs as well as for divisible multiple graphs in the case when multiplicity k ≥ 3. The corresponding optimization problems are NP-hard.

Tadpole domination in duplicated graphs

In this paper, selected domination parameters are discussed and proved especially after expanding the graph by duplicating vertices or edges. The tadpole domination number after expansion is obtained in terms of the old tadpole domination number and the maximal path of the original graph, tadpole domination number was determined for any graph after the duplication of the order of the vertices set of G or the duplication of the size of the edges set. Determining if a graph is Hamiltonian is much more problematic. Therefore, the leading outcome in this paper is that we provide that if an expanded graph (by duplicating each vertex by an edge) has tadpole domination, then the original (initial) graph has Hamiltonian path and vice versa.

Coverage Motion Planning Using Graph in Forest Environment in Field Robot

This paper addresses the problem of using a mobile, autonomous robot to manage a forest whose trees are destined for eventual harvesting. We have been focussing a eliminate weeding operation because it is one of the hard work in the forestry works. This research proposing the computation of trajectory capable of traversing in the entire forest. The method is based on a graph whose vertices are trees located in the forest. Trees located in the forest will be treated as vertices in a graph. In the first, the initial graph is made with considering the safety of the robot. Next, editing the initial graph to be Eulerian, and finally, the Hamiltonian circuit is obtained which could be used for trajectory. By our proposed method, the trajectory of which feasible route for traversing of the entire forest would be obtained. In the experiment, we show the result of the method applying to actual artificial forest.

TO THE QUESTION OF EVALUATING CRITICITY OF ELEMENTS OF ERGATIC SYSTEM

The paper presents a scientific and methodological approach that allows to formalize the definition of the reliability of the ergatic system by determining its critical elements. The proposed approach is based on a matrix method for determining the performance of a complex ergatic system. The integral index is a numerical expression of the system performance. Calculating it for all the different variants of placing the «source» of the system operation allows you to identify the most critical elements to ensure maximum performance. The actual dependence of the value of the integral index of the system on the initial graph of its functioning is shown on a specific example.

THE HOSOYA INDEX OF GRAPHS FORMED BY A FRACTAL GRAPH

The computational complexity of the Hosoya index of a given graph is NP-Complete. Let [Formula: see text] be the graph constructed from [Formula: see text] by a triangle instead of all vertices of the initial graph [Formula: see text]. In this paper, we characterize the Hosoya index of the graph [Formula: see text]. To our surprise, it shows that the Hosoya index of [Formula: see text] is thoroughly given by the order and degrees of all the vertices of the initial graph [Formula: see text].

Bipartite Consensus of High-Order Edge Dynamics on Coopetition Multiagent Systems

The bipartite consensus of high-order edge dynamics is investigated for coopetition multiagent systems, in which the cooperative and competitive relationships among agents are characterized by positive weight and negative weight, respectively. By mapping the initial graph to a line graph, the distributed control protocol is proposed for the strongly connected, digon sign-symmetric structurally balanced line graph; and then we give sufficient conditions for the third-order multi-gent system to achieve both the bipartite consensus of edge dynamics and the final value of bipartite consensus. By transforming the coefficients of characteristic polynomial from complex domain to real number domain, the sufficient conditions for the bipartite consensus of high-order edge dynamics are also proposed, and the final values of the high-order edge dynamics on multiagent systems are obtained.

Strategic port graph rewriting: an interactive modelling framework

We present strategic port graph rewriting as a basis for the implementation of visual modelling tools. The goal is to facilitate the specification and programming tasks associated with the modelling of complex systems. A system is represented by an initial graph and a collection of graph rewrite rules, together with a user-defined strategy to control the application of rules. The traditional operators found in strategy languages for term rewriting have been adapted to deal with the more general setting of graph rewriting, and some new constructs have been included in the strategy language to deal with graph traversal and management of rewriting positions in the graph. We give a formal semantics for the language, and describe its implementation: the graph transformation and visualisation tool Porgy.