Eigenvector centrality is a linear algebra based graph invariant used in various rating systems such as webpage ratings for search engines. A generalization of the eigenvector centrality invariant is defined which is motivated by the need to design rating systems for bipartite graph models of time-sensitive and other processes. The linear algebra connection and some applications are described.

Abstract Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant each is moment. We give an explicit and efficiently computable formula for the moment of Jacobian in terms potential theory on underlying graph. show that there exists universal linear relation between moment, certain capacity tau invariant, total ...

Let G and H be graphs. The strong product GH of graphs G and H is the graph with vertex set V(G)V(H) and u=(u1, v1) is adjacent with v= (u2, v2) whenever (v1 = v2 and u1 is adjacent with u2) or (u1 = u2 and v1 is adjacent with v2) or (u1 is adjacent with u2 and v1 is adjacent with v2). In this paper, we first collect the earlier results about strong product and then we present applications of ...

Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a complicated structure determined to a large extent by the constraints and dimension of the system. As these networks are of great interest as dynamical models of biological and cognitive processes, it is useful to understand...

A factor graph of a point process is a graph whose vertices are the points of the process, and which is constructed from the process in a deterministic isometry-invariant way. We prove that the d-dimensional Poisson process has a one-ended tree as a factor graph. This implies that the Poisson points can be given an ordering isomorphic to the usual ordering of the integers in a deterministic iso...

In this paper, we examine the uniqueness (discrimination power) of a newly proposed graph invariant based on the matrix DMAX defined by Randić et al. In order to do so, we use exhaustively generated graphs instead of special graph classes such as trees only. Using these graph classes allow us to generalize the findings towards complex networks as they usually do not possess any structural const...

We study the relations of knots and links contained in a spatial graph. This is an survey article on the results about knots and links contained in a spatial graph. We do not intend to cover all results in this topic. We only treat some of them here. The set of knots and links contained in a spatial graph is a naive invariant of spatial graph. However it is of course not a complete invariant in...

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are respon...

Path-block-cycle inequalities are valid, and sometimes facet-defining, inequalities for polytopes in connection with graph partitioning problems and corresponding multicut problems. Special cases of the inequalities were introduced by De Souza & Laurent (1995) and shown to be facet-defining for the equicut polytope. Generalizations of these inequalities were shown by Ferreira et al. (1996) to b...

The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin in [ABS00] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [ABS04b], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-null...