Abstract
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.
| Language | English |
|---|---|
| Place of Publication | Tilburg |
| Publisher | CentER, Center for Economic Research |
| Number of pages | 26 |
| Volume | 2016-007 |
| Publication status | Published - 15 Feb 2016 |
Publication series
| Name | CentER Discussion Paper |
|---|---|
| Volume | 2016-007 |
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Keywords
- binomal coefficient
- Pascal's triangle
- graph
- Markov process
- centrality measure
Cite this
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Generalization of Binomial Coefficients to Numbers on the Nodes of Graphs. / Khmelnitskaya, A.; van der Laan, G.; Talman, Dolf.
Tilburg : CentER, Center for Economic Research, 2016. (CentER Discussion Paper; Vol. 2016-007).Research output: Working paper › Discussion paper › Other research output
TY - UNPB
T1 - Generalization of Binomial Coefficients to Numbers on the Nodes of Graphs
AU - Khmelnitskaya, A.
AU - van der Laan, G.
AU - Talman, Dolf
PY - 2016/2/15
Y1 - 2016/2/15
N2 - 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'striangle 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.
AB - 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'striangle 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.
KW - binomal coefficient
KW - Pascal's triangle
KW - graph
KW - Markov process
KW - centrality measure
M3 - Discussion paper
VL - 2016-007
T3 - CentER Discussion Paper
BT - Generalization of Binomial Coefficients to Numbers on the Nodes of Graphs
PB - CentER, Center for Economic Research
CY - Tilburg
ER -