Number Of Triangles In Graph Research Articles

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

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 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.

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.

Using shortcut edges to maximize the number of triangles in graphs

In this paper, we consider the following problem: given an undirected graph G=(V,E) and an integer k, find I⊆V2 with |I|≤k in such a way that G′=(V,E∪I) has the maximum number of triangles (a cycle of length 3). We first prove that this problem is NP-hard and then give an approximation algorithm for it.

Counting triangles in real-world networks using projections

Triangle counting is an important problem in graph mining. Two frequently used metrics in complex network analysis that require the count of triangles are the clustering coefficients and the transitivity ratio of the graph. Triangles have been used successfully in several real-world applications, such as detection of spamming activity, uncovering the hidden thematic structure of the web and link recommendation in online social networks. Furthermore, the count of triangles is a frequently used network statistic in exponential random graph models. However, counting the number of triangles in a graph is computationally expensive. In this paper, we propose the EigenTriangle and EigenTriangleLocal algorithms to estimate the number of triangles in a graph. The efficiency of our algorithms is based on the special spectral properties of real-world networks, which allow us to approximate accurately the number of triangles. We verify the efficacy of our method experimentally in almost 160 experiments using several Web Graphs, social, co-authorship, information, and Internet networks where we obtain significant speedups with respect to a straightforward triangle counting algorithm. Furthermore, we propose an algorithm based on Fast SVD which allows us to apply the core idea of the EigenTriangle algorithm on graphs which do not fit in the main memory. The main idea is a simple node-sampling process according to which node i is selected with probability $${\frac{d_i}{2m}}$$ where d i is the degree of node i and m is the total number of edges in the graph. Our theoretical contributions also include a theorem that gives a closed formula for the number of triangles in Kronecker graphs, a model of networks which mimics several properties of real-world networks.

Pentagons vs. triangles

We study the maximum number of triangles in graphs with no cycle of length 5 and analogously, the maximum number of edges in 3-uniform hypergraphs with no cycle of length 5.

Clustering of spectra and fractals of regular graphs

We exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices. The structure has a fractal threadlike appearance. Points with coordinates given by the mean and variance of the exponentials of graph eigenvalues cluster around a line segment that we call a filar. Zooming-in reveals that this cluster splits into smaller segments (filars) labeled by the number of triangles in graphs. Further zooming-in shows that the smaller filars split into subfilars labeled by the number of quadrangles in graphs, etc. We call this fractal structure, discovered in a numerical experiment, a multifilar structure. We also provide a mathematical explanation of this phenomenon based on the Ihara–Selberg trace formula, and compute the coordinates and slopes of all filars in terms of Bessel functions of the first kind.

On the Maximum Number of Triangles in Wheel-Free Graphs

Gallai [1] raised the question of determiningt(n), the maximum number of triangles in graphs ofnvertices with acyclic neighborhoods. Here we disprove his conjecture (t(n) ~ n2/8) by exhibiting graphs having n2/7.5 triangles. We improve the upper bound [11] of (n2−n)/6 tot(n) ≤;n2/7.02 +O(n). For regular graphs, we further decrease this bound ton2/7.75 +O(n).