Eigenvectors of adjacency matrices are useful as measures of centrality or of status. However, they are misapplied to asymmetric networks in which some positions are unchosen. For these networks, an alternative measure of centrality is suggested that equals an eigenvector when eigenvectors can be used and provides meaningfully comparable results when they cannot. © 2001 Elsevier Science B.V. Al...

Extending to r > 1 a formula of the authors, we compute the expected reflection distance of a product of t random reflections in the complex reflection group G(r, 1, n). The result relies on an explicit decomposition of the reflection distance function into irreducible G(r, 1, n)-characters and on the eigenvalues of certain adjacency matrices.

Networks of social relations can be represented by graphs and socioor adjacency-matrices and their structure can be analyzed using different concepts, one of them called centrality. We will provide a new formalization of a “node-centrality” which leads to some properties a measure of centrality has to satisfy. These properties allow to test given measures, for example measures based on degree, ...

We prove that counting copies of any graph F in another graph G can be achieved using basic matrix operations on the adjacency matrix of G. Moreover, the resulting algorithm is competitive for medium-sized F : our algorithm recovers the best known complexity for rooted 6-clique counting and improves on the best known for 9-cycle counting. Underpinning our proofs is the new result that, for a ge...

Reducing the NP-problems to the convex/linear analysis on the Birkhoff polytope. Introduction Since the classical works of J. Edmonds [2, and others], linear modeling became a common technique in combinatorial optimization [8, 9, 13, 14, 15, 16, and others]. Often, the linear models are expressed with some constrains on the incidence vector. The major benefit of this approach is the symmetry of...

We describe a statistical approach for modeling agreements and disagreements in conversational interaction. Our approach first identifies adjacency pairs using maximum entropy ranking based on a set of lexical, durational, and structural features that look both forward and backward in the discourse. We then classify utterances as agreement or disagreement using these adjacency pairs and feature...

The main goal of this paper is to estimate the magnitude of the second largest eigenvalue in absolute value, 2 , of (the adjacency matrix of) a random d-regular graph, G. In order to do so, we study the probability that a random walk on a random graph returns to its originating vertex at the k-th step, for various values of k. Our main theorem about eigenvalues is that E fj 2 (G)j m g 2 p 2d ? ...

Let G be a connected k–regular bipartite graph with bipartition V (G) = X ∪Y and adjacency matrix A. We say G is det–extremal if per(A) = |det(A)|. Det–extremal k–regular bipartite graphs exist only for k = 2 or 3. McCuaig has characterized the det–extremal 3–connected cubic bipartite graphs. We extend McCuaig’s result by determining the structure of det–extremal cubic bipartite graphs of conne...

Motivation. Novel carbon allotropes, with finite molecular structure, including spherical fullerenes are nowadays currently produced and investigated. These compounds have beautiful architectures and show unusual properties that are very promising for the development of nanotechnologies. The Kekulé structure count and permanent of the adjacency matrix are computed for these molecules. Method. A...

A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n 2 ). In this paper we discuss the ...