Resistance distance in complete [formula omitted]-partite graphs

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.

On the resistance distance and Kirchhoff index of a linear hexagonal (cylinder) chain

Extremal values on the Kirchhoff index of the line graph of trees

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

Resistance Distance in Potting Networks

Partitions of graphs and multigraphs under degree constraints

We develop several parallel algorithms for shortest distance queries in planar graphs that use graph partitioning in the preprocessing phase to precompute and store distances between selected pairs of vertices. In the query phase, given a pair of arbitrary vertices v and w, the stored information is used to find the distance between v and w fast. The algorithms are implemented and tested on a high performance cluster with upto 256 16-core CPUs and their performances are analyzed and compared.

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.

In this thesis we contribute with new theoretical results and algorithms to the research area related to cycles and paths in (directed) graphs. In Chapter 2, we improve a classical result of Woodall for Hamiltonicity in digraphs, involving the sum of the in-degree and out-degree of two vertices for which there is no arc from one to another, by characterizing the exceptional digraphs for the weakened condition. In Chapter 3, we introduce and study the concept of path extendability in digraphs. Several degree and extremal conditions for path extendability are presented, and path extendability of regular tournaments is settled. Some open problems are raised. In Chapter 4, we study cycle extendability in bipartite tournaments. We prove that in bipartite tournaments, Hamiltonicity, pancyclicity and cycle extendability are equivalent properties, with only one exceptional class that can be clearly described. We propose problems for further consideration involving the equivalence of these properties in the more general class of multipartite tournaments. In Chapter 5, we calculate the number of 2-paths between every vertex pair, a concept that one often encounters in conditions for cycle and path properties, in random oriented graphs and random tournaments. Based on our enumeration, we obtain several results related to path extendability in tournaments. In Chapter 6, we study computational problems in the context of cycle partitions and clique partitions of edge-colored graphs, which can be viewed as a generalization of digraphs. The results of this chapter deal with computational complexity and algorithmic aspects. Some new methods and tools are developed. In Chapter 5, the number of 2-paths is used in conditions or as a tool for establishing properties related to cycles and paths in tournaments. In Chapter 4, we define the concept of the in-out graph of a digraph, which is used to construct and analyze the structure of cycle factors of two intersecting cycles. To establish a degree condition for path extendability in Chapter 3, we introduce a contraction operation to relate path-extendability and cycle-extendability, so that we can use a former result of Hendry to simplify one of our proofs.

The resistance distance between two vertices of a simple connected graph G is equal to the resistance between two equivalent points on an electrical network, constructed so as to correspond to G, with each edge being replaced by a unit resistor. The Kirchhoff index of G is the sum of the resistance distances between all pairs of vertices of G. A planar graph is a nested triangle graph with 3n vertices constructed from a sequence of n triangles by joining the pairs of corresponding vertices on consecutive triangles in the sequence. In this paper, some two-vertex resistances of nested triangle network was procured by utilizing techniques from the theory of electrical networks, i.e., the series and parallel principles, the principle of substitution, the star-triangle transformation and the delta-wye transformation. And the Kirchhoff index of nested triangle network is given by the Laplacian eigenvalues.

The resistance distance was introduced by Klein and Randic as a generalization of the classical distance. The Kirchhoff index Kf (G) of a graph G is the sum of resistance distances between all unordered pairs of vertices. In this paper we determine the extremal graphs with minimal Kirchhoff index among all n-vertex graphs with k cut vertices where 1 ? k < n/2.

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

The resistance into any pair of vertices in an n-cube composed of 1 Ω resistors is one of n distinct numbers between zero and unity. Exact explicit expressions are given for the extrernal vertex pair resistance values. As n becomes large all vertex pair resistances approach the same functional value.

Traffic grooming on the path