Several topology generator algorithms have been proposed to match the heavy-tailed behavior of the real Internet AS-level Topology. Although evaluation studies have been presented for each topology generator, each of these studies assumes a different framework. Therefore, it is difficult to compare the differences between the topology generator algorithms. The objective of this paper is to fill...

Visualizing network data is applicable in domains such as biology, engineering, and social sciences. We report the results of a study comparing the effectiveness of the two primary techniques for showing network data: node-link diagrams and adjacency matrices. Specifically, an evaluation with a large number of online participants revealed statistically significant differences between the two vi...

We explore the concept of metastability in random dynamical systems, focusing on connections between random Perron–Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron–Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint ran...

The spectrum of a graph is the set of eigenvalues of its adjacency matrix. A group, together with a multiset of elements of the group, gives a Cayley graph, and a semidirect product provides a method of producing new groups. This paper compares the spectra of cyclic groups to those of their semidirect products, when the products exist. It was found that many of the interesting identities that r...

Distance energy of a graph G is a recent energy-type invariants, defined as the absolute deviation of the eigenvalues of the distance matrix of G. It is a useful molecular descriptor in QSPR modelling, as demonstrated by Consonni and Todeschini in [MATCH Commun. Math. Comput. Chem. 60 (2008), 3–14]. We describe here the distance spectrum and energy of the join-based compositions of regular grap...

Distance-regularity of a graph is in general not determined by the spectrum of the graph. The spectral excess theorem states that a connected regular graph is distance-regular if for every vertex, the number of vertices at extremal distance (the excess) equals some given expression in terms of the spectrum of the graph. This result was proved by Fiol and Garriga [From local adjacency polynomial...

An oriented graph G is a simple undirected graph G with an orientation σ, which assigns to each edge of G a direction so that G becomes a directed graph. G is called the underlying graph of G and we denote by S(G) the skew-adjacency matrix of G and its spectrum Sp(G) is called the skew-spectrum of G. In this paper, the skew spectra of two orientations of the Cartesian product of two graphs are ...

Let G be a simple graph with V (E) as its vertex (respectively, edge) set. The spectrum of G is the spectrum of its adjacency matrix. For all other definitions (or notation not given here), especially those related to graph spectra, see [1]. The purpose of this note is to provide shorter proofs of two inequalities already known in spectral graph theory. The first bounds vertex eccentricities of...

Spectral methods for clustering are now standard, and there are many toy examples in which they can be seen to yield more sensible solutions than classical schemes like vanilla k-means. A more rigorous analysis of these methods has proved elusive, however, and has so far consisted mostly of probabilistic analyses for random inputs with planted clusterings. Such an analysis, typically calls for ...