Truss Decomposition using Triangle Graphs

Recent studies have shown that social networks exhibit interesting characteristics such as community structures, i.e., vertexes can be clustered into communities that are densely connected together and loosely connected to other vertices. In order to identify communities, several definitions have been proposed that can characterize the density of connections among vertices in the networks. Dense triangle cores, also known as $k$-trusses, are subgraphs in which every edge participates at least $k-2$ triangles (a clique of size 3), exhibiting a high degree of cohesiveness among vertices. There are a number of research works that propose $k$-truss decomposition algorithms. However, existing in-memory algorithms for computing $k$-truss are inefficient for handling today’s massive networks. In this paper, we propose an efficient, yet scalable algorithm for finding $k$-trusses in a large-scale network. To this end, we propose a new structure, called triangle graph to speed up the process of finding the $k$-trusses and prove the correctness and efficiency of our method. We also evaluate the performance of the proposed algorithms through extensive experiments using real-world networks. The results of comprehensive experiments show that the proposed algorithms outperform the state-of-the-art methods by several orders of magnitudes in running time.

The triangle graph $T_6$ is not SPN

A real symmetric matrix $A$ is copositive if $x'Ax \geq 0$ for every nonnegative vector $x$. A matrix is SPN if it is a sum of a real positive semidefinite matrix and a nonnegative matrix. Every SPN matrix is copositive, but the converse does not hold for matrices of order greater than $4$. A graph $G$ is an SPN graph if every copositive matrix whose graph is $G$ is SPN. We show that the triangle graph $T_6$ is not SPN.

Generalized Line Graphs: Cartesian Products and Complexity of Recognition

Putting the concept of line graph in a more general setting, for a positive integer $k$, the $k$-line graph $L_k(G)$ of a graph $G$ has the $K_k$-subgraphs of $G$ as its vertices, and two vertices of $L_k(G)$ are adjacent if the corresponding copies of $K_k$ in $G$ share $k-1$ vertices. Then, 2-line graph is just the line graph in usual sense, whilst 3-line graph is also known as triangle graph. The $k$-anti-Gallai graph $\triangle_k(G)$ of $G$ is a specified subgraph of $L_k(G)$ in which two vertices are adjacent if the corresponding two $K_k$-subgraphs are contained in a common $K_{k+1}$-subgraph in $G$.We give a unified characterization for nontrivial connected graphs $G$ and $F$ such that the Cartesian product $G\Box F$ is a $k$-line graph. In particular for $k=3$, this answers the question of Bagga (2004), yielding the necessary and sufficient condition that $G$ is the line graph of a triangle-free graph and $F$ is a complete graph (or vice versa). We show that for any $k\ge 3$, the $k$-line graph of a connected graph $G$ is isomorphic to the line graph of $G$ if and only if $G=K_{k+2}$. Furthermore, we prove that the recognition problem of $k$-line graphs and that of $k$-anti-Gallai graphs are NP-complete for each $k\ge 3$.

REMARKS AND RIGOROUS BOUNDS RELATING TO THE DECAY Z → πγ AND ITS RELATIONSHIP TO THE ANOMALY

It is rigorously shown that the amplitude for the decay Z → πγ must fall at least as fast as [Formula: see text]. Consequently its branching ratio must be smaller than 10−4; with slightly less conservative assumptions, this limit reduces to 5 × 10−6. The anomaly, the triangle graph and the Ward identities are shown to be consistent with both these bounds and the rate for π0 → 2γ.