# Generalization of Binomial Coefficients to Numbers on the Nodes of Graphs

CentER Discussion Paper Series No. 2016-007

26 Pages Posted: 15 Feb 2016

## Anna Khmelnitskaya

St.-Petersburg State University

## Gerard van der Laan

VU University Amsterdam - Faculty of Economics and Business Administration; Tinbergen Institute - Tinbergen Institute Amsterdam (TIA)

## Dolf Talman

Tilburg University - Department of Econometrics & Operations Research

Date Written: February 15, 2016

### Abstract

The triangular array of binomial coefficients, or Pascal's triangle, is formed by starting with an apex of 1. Every row of Pascal's triangle can be seen as a line-graph, to each node of which the corresponding binomial coefficient is assigned. We show that the binomial coefficient of a node is equal to the number of ways the line-graph can be constructed when starting with this node and adding subsequently neighboring nodes one by one. Using this interpretation we generalize the sequences of binomial coefficients on each row of Pascal's triangle to so-called Pascal graph numbers assigned to the nodes of an arbitrary (connected) graph. We show that on the class of connected cycle-free graphs the Pascal graph numbers have properties that are very similar to the properties of binomial coefficients. We also show that for a given connected cycle-free graph the Pascal graph numbers, when normalized to sum up to one, are equal to the steady state probabilities of some Markov process on the nodes. Properties of the Pascal graph numbers for arbitrary connected graphs are also discussed. Because the Pascal graph number of a node in a connected graph is defined as the number of ways the graph can be constructed by a sequence of increasing connected subgraphs starting from this node, the Pascal graph numbers can be seen as a measure of centrality in the graph.

Keywords: binomial coefficient, Pascal's triangle, graph, Markov process, centrality measure

JEL Classification: C00

Suggested Citation

Khmelnitskaya, Anna and van der Laan, Gerard and Talman, Dolf, Generalization of Binomial Coefficients to Numbers on the Nodes of Graphs (February 15, 2016). CentER Discussion Paper Series No. 2016-007. Available at SSRN: https://ssrn.com/abstract=2732524 or http://dx.doi.org/10.2139/ssrn.2732524

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