Triangles In Graph Research Articles

Triangles in graphs without bipartite suspensions

Given graphs T and H, the generalized Turán number ex(n,T,H) is the maximum number of copies of T in an n-vertex graph with no copies of H. Alon and Shikhelman, using a result of Erdős, determined the asymptotics of ex(n,K3,H) when the chromatic number of H is greater than three and proved several results when H is bipartite. We consider this problem when H has chromatic number three. Even this special case for the following relatively simple three chromatic graphs appears to be challenging. The suspension Hˆ of a graph H is the graph obtained from H by adding a new vertex adjacent to all vertices of H. We give new upper and lower bounds on ex(n,K3,Hˆ) when H is a path, even cycle, or complete bipartite graph. One of the main tools we use is the triangle removal lemma, but it is unclear if much stronger statements can be proved without using the removal lemma.

Graph partitioning MapReduce-based algorithms for counting triangles in large-scale graphs

Counting number of triangles in the graph is considered a major task in many large-scale graph analytics problems such as clustering coefficient, transitivity ratio, trusses, etc. In recent years, MapReduce becomes one of the most popular and powerful frameworks for analyzing large-scale graphs in clusters of machines. In this paper, we propose two new MapReduce algorithms based on graph partitioning. The two algorithms avoid the problem of duplicate counting triangles that other algorithms suffer from. The experimental results show a high efficiency of the two algorithms in comparison with an existing algorithm, overcoming it in the execution time performance, especially in very large-scale graphs.

A Note on the Number of Triangles in Graphs Without the Suspension of a Path on Four Vertices

The suspension of the path $P_4$ consists of a $P_4$ and an additional vertex connected to each of the four vertices, and is denoted by $\hat{P_4}$. The largest number of triangles in a $\hat{P_4}$-free $n$-vertex graph is denoted by $ex(n,K_3,\hat{P_4})$. Mubayi and Mukherjee in 2020 showed that $ ex(n,K_3,\hat{P_4})= n^2/8+O(n)$. We show that for sufficiently large $n$, $ex(n,K_3,\hat{P_4})=\lfloor n^2/8\rfloor$.

On the maximum number of open triangles in graphs with the same number of vertices and edges

On the maximum number of open triangles in graphs with the same number of vertices and edges

On the Maximum Number of Open Triangles in Graphs with the Same Number of Vertices and Edges

On the Maximum Number of Open Triangles in Graphs with the Same Number of Vertices and Edges

Efficient Retrieval of Top-k Weighted Triangles on Static and Dynamic Spatial Data

Due to the proliferation of location-based services, spatial data analysis becomes more and more important. We consider graphs consisting of spatial points, where each point has edges to its nearby points and the weight of each edge is the distance between the corresponding points, as they have been receiving attention as spatial data analysis tools. We focus on triangles in such graphs and address the problem of retrieving the top-k weighted spatial triangles. This problem is computationally challenging, because the number of triangles in a graph is generally huge and enumerating all of them is not feasible. To overcome this challenge, we propose an algorithm that returns the exact result efficiently. We moreover consider two dynamic data models: (i) fully dynamic data that allow arbitrary point insertions and deletions and (ii) streaming data in a sliding-window model. They often appear in location-based services. The results of our experiments on real datasets show the efficiency of our algorithms for static and dynamic data.

Adjacency eigenvalues of graphs without short odd cycles

It is well known that spectral Turán type problem is one of the most classical problems in graph theory. In this paper, we consider the spectral Turán type problem. Let G be a graph and let G be a set of graphs, we say G is G-free if G does not contain any element of G as a subgraph. Denote by λ1 and λ2 the largest and the second largest eigenvalues of the adjacency matrix A(G) of G, respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper bound on λ12k+λ22k of n-vertex {C3,C5,…,C2k+1}-free graphs is established, where k is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most 2k+1 in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of n-vertex non-bipartite graphs with odd girth at least 2k+3, which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].

A spectral condition for odd cycles in non-bipartite graphs

Let A(G) be the adjacency matrix of a graph G and ρ(G) be its spectral radius. Given a graph H and a family F of graphs, let exsp(n,H;F)=max⁡{ρ(G)||V(G)|=n,H⊆G,Gdoes not contain any graph ofF}. Let S2k−1(Ks,t) be the graph obtained by replacing an edge of Ks,t with a copy of P2k+1, where k≥2. In this paper, we show that exsp(n,C2k+3;{C3,C5,…,C2k+1})=ρ(S2k−1(K⌈n−2k+12⌉,⌊n−2k+12⌋)) and the unique extremal graph is S2k−1(K⌈n−2k+12⌉,⌊n−2k+12⌋), which solves a question proposed in [Eigenvalues and triangles in graphs, Comb. Probab. Comput. 30 (2021) 258–270].

On Triangle Estimation Using Tripartite Independent Set Queries

Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an algorithm that approximately counts the number of triangles in a graph using only polylogarithmic queries when the number of triangles on any edge in the graph is polylogarithmically bounded. Our query oracle Tripartite Independent Set (TIS) takes three disjoint sets of vertices A, B and C as inputs, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class of group queries (Ron and Tsur ACM Trans. Comput. Theory 8(4), 15, 2016; Dell and Lapinskas 2018) and in particular is inspired by the Bipartite Independent Set (BIS) query oracle of Beame et al. (2018). We extend the algorithmic framework of Beame et al., with TIS replacing BIS, for approximately counting triangles in graphs.

SAC-COT: Sample Consensus by Sampling Compatibility Triangles in Graphs for 3-D Point Cloud Registration

Six-degree-of-freedom (6-DOF) pose estimation from feature correspondences remains a popular and robust approach for 3-D registration. However, heavy outliers that existed in the initial correspondence set pose a great challenge to this problem. This article presents a simple yet effective estimator called SAmple Consensus by sampling COmpatibility Triangles in graphs (SAC-COT) for robust 6-DOF pose estimation and 3-D registration. The key novelty is a guided three-point sampling approach. It is based on a novel correspondence sample representation, i.e., COmpatibility Triangle (COT). We first model the correspondence set as a graph with nodes connecting compatible correspondences. Then, by ranking and sampling COTs formed by ternary loops, we show that correct hypotheses can be generated in early iteration stage. Finally, the hypothesis generated by the COT yielding to the maximum consensus is the output of SAC-COT. Extensive experiments on six data sets and extensive comparisons with the state-of-the-art estimators confirm that: 1) SAC-COT can achieve accurate registrations with a few iterations and 2) SAC-COT outperforms all competitors and is ultrarobust when confronted with Gaussian noise, data decimation, holes, clutter, partial overlap, varying scales of input correspondences, and data modality variation.

Estimate the number of triangles in real-world graph streams

Estimating the number of triangles in the graph streams is the basis of data mining, which aims to design an efficient graph stream algorithm to estimate the number of triangles in graph. Real-world graph is a multi-layer graph encompassing multiple distinct types of connectivity. The state-of-the-art approaches that counting triangles mainly focus on a general graph and cannot be applied for multi-layer graph, since duplicated edges across different layers exist. In this paper, we give the concept of several triads and triangles under the multilayer network, which truly reflect the real-world network topology. And we design a new two-stage sample algorithm based on reservoir sampling and triad sampling under real-world graph streams which solve the problem of more data brought by multi-layer networks. The algorithm is also a one-pass algorithm, and it can calculate the number of all types of triangles at the same time. We analyze the expectation and variance of the estimations and show that the algorithm is unbiased and stable. Our experimental results demonstrate that algorithm has good time efficiency and accuracy.

A generalization of Tuza's conjecture

AbstractA famous conjecture of Tuza is that the minimal number of edges needed to cover all triangles in a graph is at most twice the maximal size of a set of edge‐disjoint triangles. We propose a wider setting for this conjecture. For a hypergraph let be the maximal size of a collection of edges, no two of which share or more vertices, and let be the minimal size of a collection of sets of vertices, such that every edge in contains a set from . We conjecture that the maximal ratio is attained in hypergraphs for which . This would imply, in particular, the following generalization of Tuza's conjecture: if is 3‐uniform, then . (Tuza's conjecture is the case in which is the set of all triples of vertices of triangles in any given graph.) We show that most known results on Tuza's conjecture go over to this more general setting. We also prove some general results on the ratio , and study the fractional versions and the case of ‐partite hypergraphs.

Fast Parallel Algorithms for Counting and Listing Triangles in Big Graphs

Big graphs (networks) arising in numerous application areas pose significant challengesfor graph analysts as these graphs grow to billions of nodes and edges and are prohibitively large to fit in the main memory. Finding the number of triangles in a graph is an important problem in the mining and analysis of graphs. In this article, we present two efficient MPI-based distributed memory parallel algorithms for counting triangles in big graphs. The first algorithm employs overlapping partitioning and efficient load balancing schemes to provide a very fast parallel algorithm. The algorithm scales well to networks with billions of nodes and can compute the exact number of triangles in a network with 10 billion edges in 16 minutes. The second algorithm divides the network into non-overlapping partitions leading to a space-efficient algorithm. Our results on both artificial and real-world networks demonstrate a significant space saving with this algorithm. We also present a novel approach that reduces communication cost drastically leading the algorithm to both a space- and runtime-efficient algorithm. Further, we demonstrate how our algorithms can be used to list all triangles in a graph and compute clustering coefficients of nodes. Our algorithm can also be adapted to a parallel approximation algorithm using an edge sparsification method.

Minimizing the number of 5-cycles in graphs with given edge-density

AbstractMotivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle C5. We show that every graph of order n and size $ (1 - 1/k) \left( {\matrix{n \cr 2 }} \right) $, where k ≥ 3 is an integer, contains at least $$({1 \over {10}} - {1 \over {2k}} + {1 \over {{k^2}}} - {1 \over {{k^3}}} + {2 \over {5{k^4}}}){n^5} + o({n^5})$$ copies of C5. This bound is optimal, since a matching upper bound is given by the balanced complete k-partite graph. The proof is based on the flag algebras framework. We also provide a stability result. An SDP solver is not necessary to verify our proofs.

The maximum number of induced open triangles in graphs of a given order

An open triangle is a simple, undirected graph consisting of three vertices and two edges. It is shown that the maximum number of induced open triangles in a graph on n vertices is given by $$\left( \frac{n}{2}-1\right) \lfloor \frac{n}{2}\rfloor \lceil \frac{n}{2}\rceil $$ . The maximum is achieved for the complete bipartite graph $$K_{\lfloor {n}/{2}\rfloor , \lceil {n}/{2}\rceil }$$ . The maximum expected number of open triangles in a uniform random graph on n vertices is observed to be asymptotically equivalent.

A note on the bounds of Laplacian-energy-like-invariant

The Laplacian-energy-like of a simple connected graph G is defined as LEL:=LEL(G)=∑_(i=1)^n√(μ_i ), Where μ_1 (G)≥μ_2 (G)≥⋯≥μ_n (G)=0 are the Laplacian eigenvalues of the graph G. Some upper and lower bounds for LEL are presented in this note. Moreover, throughout this work, some results related to lower bound of spectral radius of graph are obtained using the term of ΔG as the number of triangles in graph.

An improved combinatorial algorithm for Boolean matrix multiplication

We present a new combinatorial algorithm for triangle finding and Boolean matrix multiplication that runs in Oˆ(n3/log4⁡n) time, where the Oˆ notation suppresses poly(loglog) factors. This improves the previous best combinatorial algorithm by Chan that runs in Oˆ(n3/log3⁡n) time. Our algorithm generalizes the divide-and-conquer strategy of Chan's algorithm.Moreover, we propose a general framework for detecting triangles in graphs and computing Boolean matrix multiplication. Roughly speaking, if we can find the “easy parts” of a given instance efficiently, we can solve the whole problem faster than n3.

High Performance Exact Triangle Counting on GPUs

This paper presents a GPU implementation of the graph triangle counting operation based on the set intersection algorithm. The algorithm is implemented in four kernels optimized for different types of graphs in a code delivering performance higher than the current state-of-the-art and without preprocessing the input graph. At runtime, a lightweight heuristic is used to select the kernel to run based on the specific graph taken as input. In contrast to previous works, the presented approach takes advantage of a set intersection operation implemented via bitmaps. Moreover, the simplicity of the approach allows the code to have limited size and engineering complexity.

The minimum number of triangles in graphs of given order and size

In the 1940s and 50s, Erdős and Rademacher raised the quantitative question of determining the number of triangles one can guarantee in a graph of given order and size. This problem has garnered much attention and, in a major breakthrough, was solved asymptotically by Razborov in 2008, whose results were extended by Nikiforov and Reiher. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from one. This proves almost every case of a conjecture of Lovász and Simonovits from 1975.

Graphing trillions of triangles.

The increasing size of Big Data is often heralded but how data are transformed and represented is also profoundly important to knowledge discovery, and this is exemplified in Big Graph analytics. Much attention has been placed on the scale of the input graph but the product of a graph algorithm can be many times larger than the input. This is true for many graph problems, such as listing all triangles in a graph. Enabling scalable graph exploration for Big Graphs requires new approaches to algorithms, architectures, and visual analytics. A brief tutorial is given to aid the argument for thoughtful representation of data in the context of graph analysis. Then a new algebraic method to reduce the arithmetic operations in counting and listing triangles in graphs is introduced. Additionally, a scalable triangle listing algorithm in the MapReduce model will be presented followed by a description of the experiments with that algorithm that led to the current largest and fastest triangle listing benchmarks to date. Finally, a method for identifying triangles in new visual graph exploration technologies is proposed.