Optimal trees for functions of internal distance

On [formula omitted]-span and [formula omitted]-span of trees and full binary trees

The Wiener index is a topological index defined as the sum of distances between all pairs of vertices in a graph. It was introduced as a structural descriptor for molecular graphs of alkanes, which are trees with vertex degrees of four at the most. The terminal Wiener index is defined as the sum of distances between all pairs of pendent vertices in a graph. In this paper we investigate Wiener and terminal Wiener for graphs derived from certain operations.

The sum of distances between every pair of vertices in a graph G is called the Wiener index of G. This graph invariant was initially utilized to predict certain physico-chemical properties of organic compounds. However, the Wiener index of G does not account for any of its symmetries, which are also known to effect these physico-chemical properties. Graovac and Pisanski modified the Wiener index of G to measure the average distance each vertex is displaced under the elements of the symmetry group of G; we call this the Graovac-Pisanski (GP) distance number of G. In this article, we prove that the set of all GP distance numbers of graphs with isomorphic symmetry groups is dense in a half-line. Moreover, for each finite group Γ and each rational number q within this half-line, we present a construction for a graph whose GP distance number is q and whose symmetry group is isomorphic to Γ. This construction results in graphs whose vertex orbits are not connected; we also consider an analogous construction which ensures that all vertex orbits are connected.

The chemist Harold Wiener found 𝒲(G), the sum of distances between all pairs of vertices in a connected graph G, to be useful as a predictor of certain physical and chemical properties. The q‐analogue of 𝒲, called the Wiener polynomial 𝒲(G; q), is also useful, but it has few existing useful formulas. We will evaluate 𝒲(G; q) for certain graphs G of chemical interest. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

The Wiener index is defined as the summation of distances between all pairs of vertices in a graph or in a hypergraph. Both models—graph-theoretical and hypergraph-theoretical—are used in mathematical chemistry for quantitatively studying physical and chemical properties of classical and nonclassical organic compounds. In this paper, we consider relationships between hypertrees and trees and hypercycles and cycles with respect to their Wiener indices.

How similar are two data-cubes? In other words, the question under consideration is: given two sets of points in a multidimensional hierarchical space, what is the distance value between them? In this paper we explore various distance functions that can be used over multidimensional hierarchical spaces. We organize the discussed functions with respect to the properties of the dimension hierarchies, levels and values. In order to discover which distance functions are more suitable and meaningful to the users, we conducted two user study analysis. The first user study analysis concerns the most preferred distance function between two values of a dimension. The findings of this user study indicate that the functions that seem to fit better the user needs are characterized by the tendency to consider as closest to a point in a multidimensional space, points with the smallest shortest path with respect to the same dimension hierarchy. The second user study aimed in discovering which distance function between two data cubes, is mostly preferred by users. The two functions that drew the attention of users where (a) the summation of distances between every cell of a cube with the most similar cell of another cube and (b) the Hausdorff distance function. Overall, the former function was preferred by users than the latter; however the individual scores of the tests indicate that this advantage is rather narrow.

The distance d(u, v|G) between the vertices u and v of a simple connected graph G is the length of any shortest path in G connecting u and v. The Wiener index W(G) of G is defined as the sum of distances between all pairs of vertices of G. The edge-Wiener index of G is conceived in an analogous manner as the sum of distances between all pairs of edges of G. Two possible distances d 0(e, f|G) and d 4(e, f|G) between the edges e and f of G can be considered and according to them, the corresponding edge-Wiener indices \( {W}_{e_0}(G) \) and \( {W}_{e_4}(G) \) are defined. In this chapter, we report our recent results on computing the first and second edge-Wiener indices of some composite graphs. Results are illustrated by some interesting examples.

Recently, it has been shown that under a broad set of conditions, the commonly used distance functions will become unstable in high-dimensional data space; i.e., the distance to the farthest data point approaches the distance to the nearest data point of a given query point with increasing dimensionality. It has been shown that if dimensions are independently distributed, and normalized to have zero mean and unit variance, instability happens. In this paper, it is shown that the normalization condition is not necessary, but all appropriate moments must be finite. Furthermore, a new distance function, namely multiplicative distance, is introduced. It is theoretically proved that this function is stable for data with independent dimensions (with identical or nonidentical distribution). In contrast to usual distance functions which are based on the summation of distances over all dimensions (distance components), the multiplicative distance is based on the multiplication of distance components. Experimental results show the stability of the multiplicative distance for data with independent and correlated dimensions in the high-dimensional space and the superiority of the multiplicative distance over the norm distances for the high-dimensional data.

Local properties of geometric graphs

The expected values for the Kirchhoff indices in the random cyclooctatetraene and spiro chains

The objective of this study is to compute the Kirchhoff index and resistance distance for two classes of windmill graphs, namely the French windmill graph and the Dutch windmill graph. In this study, G is considered a simple connected graph with vertex set V (G) and edge set E(G). N is supposed to represent a network derived from G by substituting a 1-ohm resistor for each edge of G. In that case, the resistance between υ,ν ∈ V (G) is considered analogous to the resistance between two equivalent nodes in network N. We employed techniques from electrical network theory to compute the resistance distance and Kirchhoff index. The Kirchhoff index of G is the sum of the resistance distances between all pairs of vertices in G. Our computations revealed specific patterns and relationships in the resistance distances and Kirchhoff indices across different classes of windmill graphs. In addition, the Kirchhoff index and resistance distance are computed in this study for specific generalizations of these graphs. The derived equations can inspire further investigation into the resistance distance and Kirchhoff index in real-world windmill networks. Additionally, they offer a chemical framework for future research, aiding in the determination of molecular structures and characteristics.

Let G be a connected graph. The resistance distance between any two vertices of G is defined as the net effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index is the sum of resistance distances between all pairs of vertices in G. Zhou and Trinajstic (Chem Phys Lett 455(1–3):120–123, 2008) obtained a Nordhaus-Gaddum-type result for the Kirchhoff index by obtaining lower and upper bounds for the sum of the Kirchhoff index of a graph and its complement. In this paper, by making use of the Cauchy-Schwarz inequality, spectral graph theory and Foster’s formula, we give better lower and upper bounds. In particular, the lower bound turns out to be tight. Furthermore, we establish lower and upper bounds on the product of the Kirchhoff index of a graph and its complement.

The minimum Kirchhoff index of phenylene chains

Neighborhood unions and hamiltonian properties in graphs

The resistance distance is a novel distance function on a graph proposed by Klein and Randić [D.J. Klein and M. Randić, Resistance distance, J. Math. Chem. 12 (1993), pp. 81–85]. The Kirchhoff index of a graph G is defined as the sum of resistance distances between all pairs of vertices of G. In this article, based on the result by Gutman and Mohar [I. Gutman and B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Comput. Sci. 36 (1996), pp. 982–985], we compute the Kirchhoff index of the square, 8.8.4, hexagonal and triangular lattices, respectively.