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Discussiones Mathematicae Graph Theory 33(4) (2013)
785-790
DOI: https://doi.org/10.7151/dmgt.1698
Sharp upper and lower bounds on the number of spanning trees in Cartesian product of graphs
Jernej Azarija
Department of Mathematics, University of Ljubljana Jadranska 21, 1000 Ljubljana, Slovenia |
Abstract
Let G
1 and G
2 be simple graphs and let n
1 = |V(G
1) |, m
1 = |E(G
1) |, n
2 = |V(G
2) | and m
2 = |E(G
2) |. In this paper we derive sharp upper and lower bounds for the number of spanning trees τ in the Cartesian product G
1◻G
2 of G
1 and G
2. We show that:
τ(G1◻G2) ≥ 2(n1 −1)(n2 −1)/(n1n2) ( τ(G1) n1)(n2+1)/2) ( τ(G2)n2)(n1+1)/2 |
|
and
τ(G1◻G2) ≤ τ(G1) τ(G2) [ 2m1/(n1 −1) + 2m2/(n2 −1) ](n1 −1)(n2 −1). |
|
We also characterize the graphs for which equality holds. As a by-product we derive a formula for the number of spanning trees in K
n1◻K
n2 which turns out to be n
1n1 −2n
2n2 −2(n
1+n
2)
(n1 −1)(n2 −1).
Keywords: Cartesian product graphs, spanning trees, number of spanning trees, inequality
2010 Mathematics Subject Classification: 05C76.
References
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Received 16 April 2012
Revised 29 September 2012
Accepted 1 October 2012
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