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.

Seriation is an important ordering problem which consists of finding the best ordering of a set of units whose interrelationship is defined by a bipartite graph. It has important applications in, e.g., archaeology, anthropology, psychology, and biology. This paper presents a Matlab implementation of an algorithm for spectral seriation by Atkins et al., based on the use of the Fiedler vector of the Laplacian matrix associated to the problem, which encodes the set of admissible solutions into a PQ-tree. We introduce some numerical technicalities in the original algorithm to improve its performance, and point out that the presence of a multiple Fiedler value may have a substantial influence on the computation of an approximated solution, in the presence of inconsistent data sets. Practical examples and numerical experiments show how to use the toolbox to process data sets deriving from real-world applications.

No abstract

No abstract

No abstract