相关论文: Measure Theory on Graphs

Graphs on integer points of polytopes whose edges come from a set of allowed differences are studied. It is shown that any simple graph can be embedded in that way. The minimal dimension of such a representation is the fiber dimension of…

The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order $n$ whose metric subgraphs are all paths if and only…

The Carath\'eodory theorem on the construction of a measure is generalized by replacing the outer measure with an approximation of it and generalizing the Carath\'eodory measurability. The new theorem is applied to obtain dynamically…

A space for gauge theories is defined, using projective limits as subsets of Cartesian products of homomorphisms from a lattice on the structure group. In this space, non-interacting and interacting measures are defined as well as functions…

Signed graphs have their edges labeled either as positive or negative. Here we introduce two types of signed distance matrix for signed graphs. We characterize balance in signed graphs using these matrices and we obtain explicit formulae…

We study the directed maximum common edge subgraph problem (DMCES) for the class of directed graphs that are finite, weakly connected, oriented, and simple. We use DMCES to define a metric on partially ordered sets that can be represented…

Threshold graphs are a prevalent and widely studied class of simple graphs. They have several equivalent definitions which makes them a go-to class for finding examples and counter examples when testing and learning. This versatility has…

In this paper, we study the supports of measures in multiplicative free semigroups on the positive real line and on the unit circle. We provide formulas for the density of the absolutely continuous parts of measures in these semigroups. The…

In Part I of this work we defined a generalization of the concept of effective resistance to directed graphs, and we explored some of the properties of this new definition. Here, we use the theory developed in Part I to compute effective…

Graph transformations definable in logic can be described using the notion of transductions. By understanding transductions as a basic embedding mechanism, which captures the possibility of encoding one graph in another graph by means of…

Metric algebras are metric variants of $\Sigma$-algebras. They are first introduced in the field of universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. Recently a similar notion of…

In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the…

We describe recent achievements in the theory of weight systems, which are functions on chord diagrams satisfying so-called $4$-term relations. Our main attention is devoted to constructions of weight systems. The two main sources of these…

The metric dimension of a graph is the size of the smallest set of vertices whose distances distinguish all pairs of vertices in the graph. We show that this graph invariant may be calculated by an algorithm whose running time is linear in…

The theory of graph limits is only understood to any nontrivial degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that the most important…

We use Pesin theory to study possible equilibrium measures for piecewise monotone maps of the interval. The maps may have unbounded derivative.

In this manuscript, we provide a concise review of the concept of metric dimension for both deterministic as well as random graphs. Algorithms to approximate this quantity, as well as potential applications, are also reviewed. This work has…

The paper systematically classifies rings based on the dominant metric dimensions (Ddim) of their associated CZDG, establishing consequential bounds for the Ddim of these compressed zero-divisor graphs. The authors investigate the interplay…

In this paper, we propose a family of graph partition similarity measures that take the topology of the graph into account. These graph-aware measures are alternatives to using set partition similarity measures that are not specifically…

We give upper and lower bounds for the Hausdorff dimensions for a class of graph-directed measures when its underlying directed graph is the infinite N-ary tree. These measures are different from graph-directed self-similar measures driven…