Related papers: Relations between some topological indices and the…
In this survey, we explore recent literature on finding the cores of higher graphs using geometric and topological means. We study graphs, hypergraphs, and simplicial complexes, all of which are models of higher graphs. We study the notion…
In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…
An atom $a$ of a molecular graph $G$ uniquely determines (resolves) a pair $(a_1,a_2)$ of atoms of $G$ if the distance between $a$ and $a_1$ is different from the distance between $a$ and $a_2$. In this paper, we quantify the involvement of…
We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of…
A concrete analysis of the general properties and numerical characteristics of different atomic and nuclear shell systems and subnuclear particles is carried out on the base of the solution scheme for an introduced in part I physical graph…
Past approaches for statistical shape analysis of objects have focused mainly on objects within the same topological classes, e.g., scalar functions, Euclidean curves, or surfaces, etc. For objects that differ in more complex ways, the…
Structural features are important features in a geometrical graph. Although there are some correlation analysis of features based on covariance, there is no relevant research on structural feature correlation analysis with graph neural…
We improve recent results relating graph eigenvalues to other graph parameters like girth, domination number, and minimum degree.
Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways…
The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist.
Systems as diverse as mechanical structures assembled from elastic components, and photonic metamaterials enjoy a common geometrical feature: a sublattice symmetry. This property realizes a chiral symmetry first introduced to characterize a…
The vertex $k$-partiteness $v_k(G)$ of graph $G$ is defined as the fewest number of vertices whose deletion from $G$ yields a $k$-partite graph. In this paper, we introduce two concepts: monotonic decreasing topological index and monotonic…
The atom-bond connectivity (ABC) index is a degree-based molecular structure descriptor that can be used for modelling thermodynamic properties of organic chemical compounds. Motivated by its applicable potential, a series of investigations…
In data analysis, there is a strong demand for graph metrics that differ from the classical shortest path and resistance distances. Recently, several new classes of graph metrics have been proposed. This paper presents some of them…
We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the…
J. Przytycki has established a connection between the Hochschild homology of an algebra $A$ and the chromatic graph homology of a polygon graph with coefficients in $A$. In general the chromatic graph homology is not defined in the case…
Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be two graphs with disjoint vertex sets $V_1$ and $V_2$. Let $u_1 \in V_1$ and $u_2 \in V_2$. A splice of $G_1$ and $G_2$ by vertices $u_1$ and $u_2$, $\mathcal{S}(G_1,G_2;u_1,u_2)$, is defined by…
Let $G$ be a graph with order $n(G)$, size $m(G)$, first Zagreb index $M_1(G)$, and second Zagreb index $M_2(G)$. More than twenty years ago, it was conjectured that $\frac{M_1(G)}{n(G)} \leq \frac{M_2(G)}{m(G)}$. Later, Hansen and…
This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends,…