2 Networks 2 2.1 graphNEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Adjacency Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Extract the Largest Connected Subgraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Enumerate Edge-Deleted Subgraphs . . . . . . . . . . . . . . ...

A controllable network can be driven from any initial state to any desired state using driver nodes. A set of driver nodes to control a network is not unique. It is important to characterize these driver nodes and select the right driver nodes. The work discusses theory and algorithms to select driver node such that largest region of attraction can be obtained considering limited capacity of dr...

Adjacency and co-occurence are two well separated notions: even if they are the same for graphs, they start to be two different notions for uniform hypergraphs. After having done the difference between the two notions, this paper contributes in the definition of a co-occurence tensor reflecting the general hypergraph structure. It is a challenging issue that can have many applications if proper...

First, we give summary of the present values of CKM matrix elements. Then, we discuss whether CKM matrix is unitary or not, and how we can find out if it is not unitary.

Several algorithms, namely CubeMiner, Trias, and DataPeeler, have been recently proposed to mine closed patterns in ternary relations. We consider here the specific context where a ternary relation denotes the value of a graph adjacency matrix at different timestamps. Then, we discuss the constraint-based extraction of patterns in such dynamic graphs. We formalize the concept of δ-contiguous cl...

Tutte proved that, if two graphs, both with more than two vertices, have the same collection of vertex-deleted subgraphs, then the determinants of the two corresponding adjacency matrices are the same. In this paper, we give a geometric proof of Tutte’s theorem using vectors and angles. We further study the lowest eigenspaces of these adjacency matrices.

For integers n ≥ 1, k ≥ 0, and k ≤ n, the graph Γn has vertices the 2 vectors of F n 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular Γn is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k = 1, 2, 3, following results obtained for the binary codes of the n-cube ...

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and rank(G) be the rank of the adjacency matrix of G. In this paper we characterize all graphs with E(G) = rank(G). Among other results we show that apart from a few families of graphs, E(G) ≥ 2max(χ(G), n − χ(G)), where n is the number of vertices of G, G ...

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. Akbari, Cameron, and Khosrovshahi conjectured that the number of vertices of every reduced graph of rank r is at most m(r) = 2 − 2 if r is even and m(r) = 5 · 2(r−3)/2 − 2 if r is odd. In this article, we prove that i...

We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erdős-Rényi graphs. For the Erdős-Rényi graph G(n, d/n), our results imply that the smallest and second-largest eigenvalues of the adjacency matrix converge to the edges of the support of the asymptotic eigenvalue distribution provided that d log n. Toget...