Balanced Graph Research Articles

Linear instability criterion for the Korteweg–de Vries equation on metric star graphs

The aim of this work is to establish a novel linear instability criterion for the Korteweg–de Vries (KdV) model on metric graphs. In the case of balanced graphs with a structure represented by a finite collection of semi-infinite edges and with boundary condition of δ-type interaction at the graph-vertex, we show that the continuous tail and bump profiles are linearly unstable. In this case, the use of the analytic perturbation theory of operators as well as the extension theory of symmetric operators is fundamental in our stability analysis. The arguments showed in this investigation have prospects in the study of the instability of stationary waves solutions for nonlinear evolution equations on metric graph.

Distance Based Topological Indices And Regular Graphs

In this article, we are using the regular graph of even number of vertices and computing the distance balanced graphs. First we take a graph for satisfying regular definition and then we compute the Mostar index of that particular graph. If the Mostar index of that particular graph is zero, then the graph is said to be a distance balanced graph. So we discuss first distance balanced graph. Suppose if we delete one edge in that particular graph, that is non-regular graph, we can verify the balanced graph is whether distance balanced graph or not. We discuss and compute the Mostar index of certain regular and non-regular graphs are balanced distance or not. Finally we see few theorems are related in this topic. So in this paper, we study some distance based topological indices for regular graphs and also cubic graphs.

Effective partitioning mechanisms for time-evolving graphs in the Flink system

Graphs are suitable data structures for expressing the relationship between different types of data. With a continuous increase in the graph size, using suitable methods to divide graphs and parallelize the processing load becomes crucial. Balanced graph partitioning has been extensively studied for static and streaming graphs. However, for a time-evolving graph (TEG), whose size and structure are periodically updated, related partitioning methods are lacking. A straightforward approach is to capture snapshots of a TEG and adopt the partitioning methods designed for static or streaming graphs. Although feasible partitioning quality can be expected, the time overhead is high due to frequent repartitioning. This paper proposes two TEG partitioning methods, namely seed and similarity, to decrease the partitioning time. According to the experimental results, on average, seed and similarity require 29–39% of the partitioning time required by snapshot. Moreover, the proposed methods maintain reasonable partitioning quality.

Trust: Triangle Counting Reloaded on GPUs

Triangle counting is a building block for a wide range of graph applications. Traditional wisdom suggests that i) hashing is not suitable for triangle counting, ii) edge-centric triangle counting beats vertex-centric design, and iii) communication-free and workload balanced graph partitioning is a grand challenge for triangle counting. On the contrary, we advocate that i) hashing can help the key operations for scalable triangle counting on Graphics Processing Units (GPUs), i.e., list intersection and graph partitioning, ii)vertex-centric design reduces both hash table construction cost and memory consumption, which is limited on GPUs. In addition, iii) we exploit graph and workload collaborative, and hashing-based 2D partitioning to scale vertex-centric triangle counting over 1,000 GPUswith sustained scalability. In this work, we present TRUST which performs triangle counting with the hash operation and vertex-centric mechanism at the core. To the best of our knowledge, TRUSTis the first work that achieves over one trillion Traversed Edges Per Second (TEPS) rate for triangle counting.

On some applications related with algebraic structures through different well known graphs

Lot of achievements have been done to comprehend the latest phenomena of our life by discussing the interrelationship of two boughs, graph theory and algebra, of mathematics. In the modern time, it seems to be of great importance to study about eminent, extensive and certifiable applications through well known tools of graph theory. Combinatorial commutative algebra, moderately new, quickly creating numerical tool and foremost tendril of mathematics, has been used exhaustively to solve a list of open problems in algebra under the shadow of graphs. It can be helpful, instead of algebraically, to find the metric dimension of zero-divisor graphs, NP-complete problem, related with finite mathematical structures. In this article we shall discuss some graphs, zero-divisor graphs of finite Boolean rings and balanced bipartite graphs of conjugacy closed loops, and their associated applications.

Fixed-time consensus tracking for nonlinear stochastically disturbed multi-agent systems via discontinuous protocols

This paper studies the fixed-time consensus tracking (FixCT) problem for nonlinear stochastically disturbed multi-agent systems (MASs) in detail balanced graphs. Two novel and simple discontinuous control protocols are developed to facilitate consensus tracking in a fixed time. By using differential inclusions and the stochastic stability theory, rigorous consensus tracking proofs are provided; moreover, upper bounds of the settling time irrelevant to initial conditions are explicitly estimated. A numerical example is presented to corroborate the effectiveness of the proposed protocols.

Optimal Wirelength of Balanced Complete Multipartite Graphs onto Cartesian Product of {Path, Cycle} and Trees

In any interconnection network, task allocation plays a major role in the processor speed as fair distribution leads to enhanced performance. Complete multipartite networks serve well for this purpose as the task can be split into different partites which improves the degree of reliability of the network. Such an allocation process in the network can be done by means of graph embedding. The optimal wirelength of a graph embedding helps in the distribution of deterministic algorithms from the guest graph to other host graphs in order to incorporate its unique deterministic properties on that chosen graph. In this paper, we propose an algorithm to compute the optimal wirelength of balanced complete multipartite graphs onto the Cartesian product of trees with path and cycle. Moreover, we derive the closed formulae for wirelengths in specific trees like (1-rooted) complete binary tree and sibling graphs.

Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

A graph admitting a perfect matching has the Perfect–Matching–Hamiltonian property (for short the PMH–property) if each of its perfect matchings can be extended to a hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH–property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, (iii) a balanced complete bipartite graph with at least 100 vertices, or (iv) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.

EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2 Õ( n 2/3 ) for M AXIMUM C LIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for M AX C LIQUE on disk and unit ball graphs. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3 , a maximum subset of points with diameter at most some fixed value. In stark contrast, M AXIMUM C LIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2 n 1−ɛ , unless the Exponential Time Hypothesis fails.

Biholes in balanced bipartite graphs

A bihole in a bipartite graph $G$ with partite sets $A$ and $B$ is an independent set $I$ in $G$ with $|I\cap A|=|I\cap B|$. We prove lower bounds on the largest order of biholes in balanced bipartite graphs subject to conditions involving the vertex degrees and the average degree.

A construction of balanced degree-magic graphs

A graph G is called degree-magic if it admits a labelling of the edges by integers 1,2,...,|E(G)| such that the sum of the labels of the edges incident with any vertex v is equal to (1+|E(G)|)deg(v)/2. Degree-magic graphs extend supermagic regular graphs. In this paper, a new construction of balanced degree-magic graphs is introduced.

ANALYSIS OF THE BRAZILIAN STOCK MARKET THROUGH GRAPH CENTRALITY MEASURES

ABSTRACT This article aims to analyze the shares that make up the Brazilian IBRX100 index, verifying which sectors had the greatest influence on the Stock Exchange in 2018, 2019, and 2020. For this purpose, the theory of graph centrality measures was used to discover the most central shares. A balance analysis of the graphs was also performed, since balanced graphs are more stable, generating a more predictable stock portfolio. This study may help investors to compose a safer stock portfolio and identify which stocks are most correlated with each other. The most central shares can aid in perceiving stock market trends.

Bipartite containment of descriptor multi‐agent systems via an observer‐based approach

This study considers the bipartite containment problem for a class of descriptor multi-agent systems (MASs) under a structurally balanced graph. A new bipartite containment control protocol acting on each follower is proposed on the basis of designing a singular observer, where relative output information is utilised. By employing the stability theory of descriptor systems and multiple Lyapunov function technique, some sufficient criteria are derived to guarantee observer-based bipartite containment of descriptor MASs. The feasibility of obtained theoretical results is finally verified via performing simulations.

An Effective 2-Dimension Graph Partitioning for Work Stealing Assisted Graph Processing on Multi-FPGAs

Multi-FPGA architectures have gained great interests in accelerating large-scale graph processing with great potential on high throughput and energy efficiency. As a beneficial complement, work stealing functions effectively to balance the computational workload on different FPGAs dynamically. Unfortunately, existing graph partitioning schemes originally designed in distributed settings potentially mismatch the work stealing-enabled multi-FPGA situations, where the computation is balanced while the communication overhead is unprecedentedly significant. In this paper, we present a 2-dimension balanced graph partitioning for work stealing assisted graph systems on multi-FPGAs, which can reduce communication overhead while preserving the optimal performance of work stealing. Our approach is novel by 1) exploring the tradeoff between load balance dimension and communication dimension in work-stealing-enabled graph processing system for the optimal performance, and 2) optimizing the memory access sequences to improve the granularity of graph partitioning for high-throughput graph analytics. Our experimental results show that our approach achieves 1.63x <inline-formula><tex-math notation="LaTeX">$\sim$</tex-math></inline-formula> 2.56x speedups compared with state-of-the-art FPGA-based graph processing systems.

Programming Strategies for Irregular Algorithms on the Emu Chick

The Emu Chick prototype implements migratory memory-side processing in a novel hardware system. Rather than transferring large amounts of data across the system interconnect, the Emu Chick moves lightweight thread contexts to near-memory cores before the beginning of each remote memory read. Previous work has characterized the performance of the Chick prototype in terms of memory bandwidth and programming differences from more typical, non-migratory platforms, but there has not yet been an analysis of algorithms on this system. This work evaluates irregular algorithms that could benefit from the lightweight, memory-side processing of the Chick and demonstrates techniques and optimization strategies for achieving performance in sparse matrix-vector multiply operation (SpMV), breadth-first search (BFS), and graph alignment across up to eight distributed nodes encompassing 64 nodelets in the Chick system. We also define and justify relative metrics to compare prototype FPGA-based hardware with established ASIC architectures. The Chick currently supports up to 68x scaling for graph alignment, 80 MTEPS for BFS on balanced graphs, and 50% of measured STREAM bandwidth for SpMV.

Extremal problems on distance spectra of graphs

Motivated by recent progresses on spectral extremal graph theory, in this paper, we aim to investigate the existence of cycles with given length in a graph in terms of its distance spectral radius. First of all, we show that if G is a connected bipartite graph with λ1(D(G))≤λ1(D(K1,n−1)), then G contains a C4 unless G≅K1,n−1. When n is sufficiently large with respect to k, as a corollary, we show that Sk(D(G))≥2n−2k if G is a C4-free bipartite graph. Besides, we prove that Sk(D(G))≥2n−2k if G is a bipartite distance regular graph. These two results partially solve a problem proposed by Lin (2019). Secondly, we give sufficient conditions for the existence of a Hamilton cycle or Hamilton path in a balanced bipartite graph in terms of the distance spectral radius.

Better Process Mapping and Sparse Quadratic Assignment

Communication and topology-aware process mapping is a powerful approach to reduce communication time in parallel applications with known communication patterns on large, distributed memory systems. We address the problem as a quadratic assignment problem (QAP) and present algorithms to construct initial mappings of processes to processors and fast local search algorithms to further improve the mappings. By exploiting assumptions that typically hold for applications and modern supercomputer systems such as sparse communication patterns and hierarchically organized communication systems, we obtain significantly more powerful algorithms for these special QAPs. Our multilevel construction algorithms employ perfectly balanced graph partitioning techniques and exploit the given communication system hierarchy in significant ways. We present improvements to a local search algorithm of Brandfass et al. (2013) and further decrease the running time by reducing the time needed to perform swaps in the assignment as well as by carefully constraining local search neighborhoods. We also investigate different algorithms to create the communication graph that is mapped onto the processor network. Experiments indicate that our algorithms not only dramatically speed up local search but also, due to the multilevel approach, find much better solutions in practice.

Step-by-step community detection in volume-regular graphs

Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph).Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations. While the results of Becchetti et al. apply to perfectly balanced partitions and/or graphs that exhibit very strong forms of regularity, we extend their approach to graphs containing a hidden k partition and characterized by a milder form of volume-regularity. We show that the class of k-volume regular graphs is the largest class of undirected (possibly weighted) graphs whose transition matrix admits k “stepwise” eigenvectors (i.e., vectors that have constant entries over the components corresponding to the same set of the hidden partition). To obtain this result, we highlight a connection between volume regularity and lumpability of Markov chains. Moreover, we prove that if the stepwise eigenvectors are those associated to the first k largest eigenvalues of the transition matrix of a random walk on the graph and the gap between the k-th and the (k+1)-th eigenvalues is sufficiently large, the Averaging dynamics of Becchetti et al. recovers the underlying community structure of the graph in logarithmic time, with high probability.

Bipartite Independent Number and Hamilton-Biconnectedness of Bipartite Graphs

Let G be a balanced bipartite graph with bipartite sets X, Y. We say that G is Hamilton-biconnected if there is a Hamilton path connecting any vertex in X and any vertex in Y. We define the bipartite independent number $$\alpha ^o_B(G)$$ to be the maximum integer $$\alpha $$ such that for any integer partition $$\alpha =s+t$$ , G has an independent set formed by s vertices in X and t vertices in Y. In this paper we show that if $$\alpha ^o_B(G)\le \delta (G)$$ then G is Hamilton-biconnected, unless G has a special construction.

Unit Ball Graphs on Geodesic Spaces

Consider finitely many points in a geodesic space. If the distance of two points is less than a fixed threshold, then we regard these two points as "near". Connecting near points with edges, we obtain a simple graph on the points, which is called a unit ball graph. If the space is the real line, then it is known as a unit interval graph. Unit ball graphs on a geodesic space describe geometric characteristics of the space in terms of graphs. In this article, we show that every unit ball graph on a geodesic space is (strongly) chordal if and only if the space is an $ \mathbb{R} $-tree and that every unit ball graph on a geodesic space is (claw, net)-free if and only if the space is a connected manifold of dimension at most $ 1 $. As a corollary, we prove that the collection of unit ball graphs essentially characterizes the real line and the unit circle.