Bounds of Fibonacci-sum graph for some distancebased topological indices and an application on BFS algorithm

Purpose of research. The research presented in this article is aimed at improving the speed of finding a path for the movement route of robots. The scientific novelty is the obtained correlation of time and field size.Methods. To find the path in the maze, the depth-first search and breadth-first search algorithms were used, the basis of which is the cyclic processing of adjacent previously unvisited graph vertices. Performance is estimated in terms of the speed of program code execution on prepared samples. Scientific novelty was obtained by studying the influence of map sizes on the performance of depth-first and breadth-first search algorithms.Results. A software implementation of breadth-first and depth-first search algorithms has been developed. The article provides a more detailed description of the breadth-first search algorithm in the form of pseudo and program codes, which are based on the while loop, where the queue of checked graph vertices is processed. Based on the evaluation of the speed of the found path, it was concluded that the breadth-first search is not the fastest. Based on the assessment of the influence of various factors on the speed of the algorithm, it was concluded that an increase in the size of the field, a decrease in the number of obstacles and a distance between the starting and final points increases the execution time of the algorithm.Conclusion. The breadth-first search algorithm and its software implementation were presented. In the course of experimental studies, it was found that this algorithm is not the fastest in time, but in all tests, it found the shortest path. The correlation ta = f(w, h) was also obtained for the prepared samples of the desired field, which is expressed as the dependence of the algorithm execution time on the length and width of the field. And we can conclude that it is applicable for finding the movement path of robots, since it always finds the shortest path.

There are many large-scale graphs in real world such as Web graphs and social graphs. The interest in large-scale graph analysis is growing in recent years. Breadth-First Search (BFS) is one of the most fundamental graph algorithms used as a component of many graph algorithms. Our new method for distributed parallel BFS can compute BFS for one trillion vertices graph within half a second, using large supercomputers such as the K-Computer. By the use of our proposed algorithm, the K-Computer was ranked 1st in Graph500 using all the 82,944 nodes available on June and November 2015 and June 2016 38,621.4 GTEPS. Based on the hybrid BFS algorithm by Beamer (Proceedings of the 2013 IEEE 27th International Symposium on Parallel and Distributed Processing Workshops and PhD Forum, IPDPSW ’13, IEEE Computer Society, Washington, 2013), we devise sets of optimizations for scaling to extreme number of nodes, including a new efficient graph data structure and several optimization techniques such as vertex reordering and load balancing. Our performance evaluation on K-Computer shows that our new BFS is 3.19 times faster on 30,720 nodes than the base version using the previously known best techniques.

The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = ∑(n eu (e|G)+ n ev (e|G)), where n eu (e|G) is the number of edges of G lying closer to u than to v, n ev (e|G) is the number of edges of G lying closer to v than to u and summation goes over all edges of G. In this paper, we first compute the PI index of a class of pericondensed benzenoid graphs consisting of n rows, n ≤ 3, of hexagons of various lengths. Finally, we prove that for any connected graph G with exactly m edges, PI(G) ≤ m(m-1) with equality if and only if G is an acyclic graph or a cycle of odd length. 1 . Here, we consider a new topological index, named the Padmakar-Ivan index, which is abbreviated as the PI index 2-17 . This newly proposed topological index, differ from the Wiener index 18 , the oldest topological index for acyclic (tree) molecules. We now describe some notations which will be adhered to throughout. Benzenoid systems (graph representations of benzenoid hydrocarbons) are defined as finite connected plane graphs with no cut-vertices, in which all interior regions are mutually congruent regular hexagons. More details on this important class of molecular graphs can be found in the book of Gutman and Cyvin 19 and in the references cited therein. Let G be a simple molecular graph without directed or multiple edges and without loops, the vertex and edge-shapes of which are represented by V(G) and E(G), respectively. The graph G is said to be connected if for every pair of vertices x and y in V(G) there exists a path between x and y. In this paper we only consider connected graphs. If e is an edge of G, connecting the vertices u and v then we write e=uv. The number of vertices of G is denoted by n. The distance between a pair of vertices u and w of G is denoted by d(u,w). We now define the PI index of a graph G. To do this, suppose that e = uv and introduce the quantities neu(e|G) and nev(e|G). neu(e|G) is the number of edges lying closer to vertex u than to vertex v, and nev(e|G) is the number of edges lying closer to vertex v than to vertex u. Then PI(G) = ∑(neu(e|G) + nev(e|G)), where the summation goes over all edges of G. Edges equidistant from both ends of the edge e = uv are not counted and the number of such edges is denoted by N(e). To clarify this, for every vertex u and any edge f = zw of graph G, we define d(f,u) = Min{d(u,w),d(u,z)}. Then f is equidistant from both ends of the edge e = uv if d(f,u) = d(f,v). In a series of papers, Khadikar and coauthors 2-17 defined and then computed the PI index of some chemical graphs. The present author 20 computed the PI index of a zig-zag polyhex nanotube. In this paper we continue this study to prove an important result concerning the PI index and find an exact expression for the PI index of some other chemical graphs. Our notation is standard and mainly taken from the literature. 21,22

The end-vertex problem for a search algorithm asks whether a vertex of the input graph is the last visited vertex of an execution of that search algorithm. We consider the end-vertex problem restricted to AT-free bigraphs for various search algorithms: Breadth-First Search (BFS), Lexicographic Breadth-First Search (LBFS), Depth-First Search (DFS), and Maximal Neighbourhood Search (MNS). Deciding whether a vertex of a graph is the end-vertex of any of these search algorithms is NP-complete in general. We show that we can decide whether a vertex is an end-vertex of BFS or MNS in polynomial time on AT-free bigraphs. Additionally, we show that we can decide whether a vertex is an end-vertex of DFS or LBFS in linear time on AT-free bigraphs; this improves the LBFS end-vertex complexity on this class of graphs.

Centrality is a family of metrics for characterizing the importance of a vertex in a graph. Although a large number of centrality metrics have been proposed, a majority of them ignores uncertainty in graph data. In this article, we formulate closeness centrality on uncertain graphs and define the batch closeness centrality evaluation problem that computes the closeness centrality of a subset of vertices in an uncertain graph. We develop three algorithms, MS-BCC , MG-BCC, and MGMS-BCC , based on sampling to approximate the closeness centrality of the specified vertices. All these algorithms require to perform breadth-first searches (BFS) starting from the specified vertices on a large number of sampled possible worlds of the uncertain graph. To improve the efficiency of the algorithms, we exploit operation-level parallelism of the BFS traversals and simultaneously execute the shared sequences of operations in the breadth-first searches. Parallelization is realized at different levels in these algorithms. The experimental results show that the proposed algorithms can efficiently and accurately approximate the closeness centrality of the given vertices. MGMS-BCC is faster than both MS-BCC and MG-BCC because it avoids more repeated executions of the shared operation sequences in the BFS traversals.

Modern applications generate massive amounts of data that is challenging to process or analyse. Graph algorithms have emerged as a solution for the analysis of such data because they can represent the entities participating in the generation of large-scale datasets in terms of vertices and their relationships in terms of edges. Graph analysis algorithms are used for finding patterns within these relationships, aiming to extract information to be further analysed. The breadth-first search (BFS) is one of the main graph search algorithms used for graph analysis and its optimisation has been widely researched using different parallel computers. However, the parallelisation of BFS has been shown to be challenging because of its inherent characteristics, including irregular memory access patterns, data dependencies and workload imbalance, that limit its scalability. This paper investigates the optimisation of the BFS on the Xeon Phi (Knights Corner), a modern parallel architecture provided with an advanced vector processor supporting the AVX-512 instruction set, using a bespoke development framework integrated with the Graph 500 benchmark. In addition, to demonstrate portability, we show results for a direct port of the algorithms to a more recent version of the Xeon Phi (Knights Landing) and to a Skylake CPU which supports most of the AVX-512 instruction set. Optimised parallel versions of two high-level algorithms for BFS were created using vectorisation, starting with the conventional top-down BFS algorithm and, building on this, a hybrid BFS algorithm. On the KNC our best implementations result in speedups of 1.37x ( top-down ) and 1.37x ( hybrid ), for a one million vertices graph, compared to the state-of-the-art. On the KNL and Skylake, the performance is higher than on KNC. In addition, we show results of our best hybrid algorithm on real-world graphs from the SNAP datasets with speedups up to 1.3x on KNC. Performance on KNL and Skylake is again higher, demonstrating the robustness and portability of our algorithm. The hybrid BFS algorithm can be further used to speed up other graph analysis algorithms and the lessons learned from vectorisation can be applied to other algorithms targeting existing and future models of the Xeon Phi and other advanced vector architectures.

We have developed a multithreaded implementation of breadth-first search (BFS) of a sparse graph using the Cilk++ extensions to C++. Our PBFS program on a single processor runs as quickly as a standar. C++ breadth-first search implementation. PBFS achieves high work-efficiency by using a novel implementation of a multiset data structure, called a bag, in place of the FIFO queue usually employed in serial breadth-first search algorithms. For a variety of benchmark input graphs whose diameters are significantly smaller than the number of vertices -- a condition met by many real-world graphs -- PBFS demonstrates good speedup with the number of processing cores.Since PBFS employs a nonconstant-time reducer -- hyperobject feature of Cilk++ -- the work inherent in a PBFS execution depends nondeterministically on how the underlying work-stealing scheduler load-balances the computation. We provide a general method for analyzing nondeterministic programs that use reducers. PBFS also is nondeterministic in that it contains benign races which affect its performance but not its correctness. Fixing these races with mutual-exclusion locks slows down PBFS empirically, but it makes the algorithm amenable to analysis. In particular, we show that for a graph G=(V,E) with diameter D and bounded out-degree, this data-race-free version of PBFS algorithm runs it time O((V+E)/P + Dlg3(V/D)) on P processors, which means that it attains near-perfect linear speedup if P

This paper studies the Maximum Internal Spanning Tree problem which is to find a spanning tree with the maximum number of internal vertices on a graph. We prove that the problem can be solved in polynomial time on interval graphs. The idea is based on the observation that the number of internal vertices in a maximum internal spanning tree is at most one less than the number of edges in a maximum path cover on any graph. On an interval graph, we present an \(O(n^{2})\)-algorithm to find a spanning tree in which the number of internal vertices is exactly one less than the number of edges in a maximum path cover of the graph, where n is the number of vertices in the interval graph.

The high-level contributions of this paper are as follows: We modify an existing branch-andbound based exact algorithm (for maximum clique size of an entire graph) to determine the maximal clique size that the individual vertices in the graph are part of. We then run this algorithm on six real-world network graphs (ranging from random networks to scale-free networks) and analyze the distribution of the maximal clique size of the vertices in these graphs. We observe five of the six real-world network graphs to exhibit a Poisson-style distribution for the maximal clique size of the vertices. We analyze the correlation between the maximal clique size and the clustering coefficient of the vertices, and find these two metrics to be poorly correlated for the real-world network graphs. Finally, we analyze the Assortativity index of the vertices of the real-world network graphs and observe the graphs to exhibit positive assortativity with respect to maximal clique size and negative assortativity with respect to node degree; nevertheless, we observe the Assortativity index of the real-world network graphs with respect to both the maximal clique size and node degree to increase with decrease in the spectral radius ratio for node degree, indicating a positive correlation between the maximal clique size and node degree.

Breadth-First Search(BFS) is one of the most fundamental graph algorithms used as a component of many graph algorithms. Our new method for distributed parallel BFS can compute BFS for one trillion vertices graph within half a second, using large supercomputers such as the K-Computer. By the use of our proposed algorithm, the K-Computer was ranked 1st in Graph500 using all the 82,944 nodes available on June and November 2015 and June 2016 38,621.4 GTEPS. Based on the hybrid-BFS algorithm by Beamer[3], we devise sets of optimizations for scaling to extreme number of nodes, including a new efficient graph data structure and optimization techniques such as vertex reordering and load balancing. Performance evaluation on the K shows our new BFS is 3.19 times faster on 30,720 nodes than the base version using the previously-known best techniques.

The Breadth-First Search (BFS) algorithm is an important building block for graph analysis of large datasets. The BFS parallelisation has been shown to be challenging because of its inherent characteristics, including irregular memory access patterns, data dependencies and workload imbalance, that limit its scalability. We investigate the optimisation and vectorisation of the hybrid BFS (a combination of top-down and bottom-up approaches for BFS) on the Xeon Phi, which has advanced vector processing capabilities. The results show that our new implementation improves by 33%, for a one million vertices graph, compared to the state-of-the-art.

The modified symmetric division deg (MSD) index of a graph G is precisely defined as MSD ( G ) = ∑ j ∼ k 1 2 d j d k + d k d j , where d j and d k represent the degrees of j and k respectively. In this paper, we present precise bounds for the modified symmetric division deg index, expressed in the relation to the minimum and maximum degrees, the order and size of the graph, the number of pendant vertices and the minimum degree of a non-pendant vertex, the forgotten topological index, the modified second Zagreb index. We determine the upper bounds for the modified symmetric division deg index of unicyclic, bicyclic and k-cyclic graphs. Moreover, upon analyzing the modified symmetric division deg index, we observe its correlation with other well-known indices and its chemical applicability on the molecular graphs of octane isomers. At the end, we find the chemical applicability of the modified symmetric division deg index on benzenoid hydrocarbons and observe that the modified symmetric division deg index has a very strong correlation with the physical properties of benzenoid hydrocarbons.

In this paper, we have proposed new windmill graph, that is Basava wheel windmill graph. The Basava wheel windmill graph \(W^{(m)}_{n+1}\) is the graph obtained by taking \(m\geq 2\) copies of the graph \(K_1+W_{n}\) for \(n\geq 4\) with a vertex \(K_1\) in common. Inspired by recent work on topological indices, proposed new degree-based topological indices namely, general \(SK_{\alpha}\) and \(SK^{\alpha}_1\) indices of a graph \(G\). We have obtained first and second Zagreb index, F-index, first and second hyper-Zagreb index, harmonic index, Randi\(\acute{c}\) index, general Randi\(\acute{c}\) index, sum connectivity index, general sum connectivity index, atom-bond connectivity index, geometric-arithmetic index, Symmetric division deg index, Sombor index, SK indices, general \(SK_{\alpha}\) and \(SK^{\alpha}_1\) indices of Basava wheel windmill graph. Further, we have computed exact values of these topological indices of chloroquine, hydroxychloroquine and remdesiver.

Solving the maximum internal spanning tree problem on interval graphs in polynomial time

For a simple connected graph G, center C(G) and periphery P(G) are subgraphs induced on vertices of G with minimum and maximum eccentricity, respectively. An n-vertex graph G is said to be an almost self-centered (ASC) graph if it contains \(n-2\) central vertices and two peripheral (diametral) vertices. An ASC graph with radius r is known as an r-ASC graph. The r-ASC index of any graph G is defined as the minimum number of new vertices, and required edges, to be introduced to G such that the resulting graph is r-ASC graph in which G is induced. For \(r=2,3\), r-ASC index of few graphs is calculated by Klavžar et al. (Acta Mathematica Sinica, 27:2343–2350, 2011 [1]), Xu et al. (J Comb Optim 36(4):1388–1410, 2017 [2]), respectively. Here we give bounds to r-ASC index of diameter two graphs and determine the exact value of this index for paths and cycles.