We determine the eccentricity of an arbitrary vertex, the average eccentricity and its standard deviation for all Sierpinski graphs ${S_p^n}$. Special cases are the graphs ${S_2^{n}}$, which are isomorphic to the state graphs of the Chinese Rings puzzle with n rings and the graphs ${S_3^{n}}$ isomorphic to the Hanoi graphs ${H_3^{n}}$ representing the Tower of Hanoi puzzle with 3 pegs and n discs.

The Average Eccentricity of Sierpiński Graphs

The status $$\sigma (u)$$
 of a vertex u is defined as the sum of the distances between u and all other vertices of a graph G. In this paper we have defined the first and second status connectivity indices of a graph G as $$S_{1}(G) = \sum \nolimits _{uv \in E(G)}[\sigma (u)+ \sigma (v)]$$
 and $$S_{2}(G) = \sum \nolimits _{uv \in E(G)}\sigma (u)\sigma (v)$$
 respectively, where E(G) denotes the edge set of G. Status connectivity indices of certain graphs are computed. Also the bounds for the status connectivity indices are determined. Further linear regression analysis of the distance based indices with the boiling points of benzenoid hydrocarbons is carried out. The linear model, based on the status index, is better than the models corresponding to the other distance based indices.

Status connectivity indices of graphs and its applications to the boiling point of benzenoid hydrocarbons

Payment channels were introduced to solve various eminent cryptocurrency scalability issues. Multiple payment channels build a network on top of a blockchain, the so-called layer 2. In this work, we analyze payment networks through the lens of network creation games. We identify betweenness and closeness centrality as central concepts regarding payment networks. We study the topologies that emerge when players act selfishly and determine the parameter space in which they constitute a Nash equilibrium. Moreover, we determine the social optima depending on the correlation of betweenness and closeness centrality. When possible, we bound the price of anarchy. We also briefly discuss the price of stability.

Ride the Lightning: The Game Theory of Payment Channels

On ( t , r ) broadcast domination numbers of grids

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 featuring the cutpoint additive distances. These include the path distances, the reliability distance, the walk distances, and the logarithmic forest distances among others. We discuss a number of connections between these and other distances.

Studying new classes of graph metrics

http://aygrt.isrj.net/UploadedData/1696.pdf

D-distance in graphs

In this paper we deal with the domination numbers of the complete grid graphs P_k 〖×P〗_n for k=15 and . Establishing this numbers for various class of graphs is the content of many papers in literature. Some of the papers which deal with the dominating number of grid graphs are [3], [4], [5]. There is no general formula for the domination number of a graph. In this paper, we use the concept of transforming the domination from a vertex in a dominating set D of a graph G = (V, E) to a vertex in V – D, where G is a simple connected graph. We give an algorithm using this transformation to obtain a domination set of a graph G.

• DOMINATION NUMBERS OF GRID GRAPHS P15×Pn

The D-distance between vertices of a graph G is obtained by considering the path lengths and as well as the degrees of vertices present on the path. The average D-distance of a connected graph is the average of the D-distance between all pairs of vertices of the graph. Similarly, the average edge D-distance is the average of D-distances between all pairs of edges in the graph. In this article we study the average edge D-distance of a graph. We find bounds for average edge D-distance which are sharp and also prove some other results.

D. Reddy Babu

Papers

D-distance in graphs

• DOMINATION NUMBERS OF GRID GRAPHS P15×Pn

Average D-Distance Between Edges of a Graph