Chained graphs and some applications

Abstract

This paper introduces the notions of chained and semi-chained graphs. The chain of a graph, when existent, refines the notion of bipartivity and conveys important structural information. Also the notion of a center vertex $$v_c$$ v c is introduced. It is a vertex, whose sum of p powers of distances to all other vertices in the graph is minimal, where the distance between a pair of vertices $$\{v_c,v\}$$ { v c , v } is measured by the minimal number of edges that have to be traversed to go from $$v_c$$ v c to v. This concept extends the definition of closeness centrality. Applications in which the center node is important include information transmission and city planning. Algorithms for the identification of approximate central nodes are provided and computed examples are presented.

Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices

“…We also discuss the computation of the Perron vector of structured matrices, somewhat related to the matrix M, and by application of the Arnoldi method to the submatrix A in (1). These particular matrices represent graphs with a chained structure that refine the notion of bipartivity [14].…”

“…The adjacency matrix for this kind of graph has a staircase structure, which generalizes the structure (1). Chained graphs have been shown to be bipartite in [14], which implies that the eigenvalues of their associated adjacency matrices appear in ± pairs. Section 3 studies the performance of the Lanczos and restarted Lanczos methods when applied to computing the Perron vector for these and other symmetric adjacency matrices.…”

“…This section describes -chained undirected graphs and the structure of their adjacency matrices. These graphs, which are particular bipartite graphs, were introduced in [14] and are defined as follows.…”

“…Chained graphs arise in various applications; see [8,14,15] and Section 5. Consider an undirected -chained graph G = (V, E ) with vertex set partitioning…”

“…Let M ∈ R n×n be the adjacency matrix of an undirected connected graph G, and let ρ and w denote the Perron root and Perron vector of M, respectively. Apply the restarted Lanczos method described above with initial vector e and without the stopping criterion (14). If the number of Lanczos steps between restarts, k, is large enough, then the computed sequence z (i) , i = 1, 2, .…”