Note: Sharp Upper and Lower Bounds on the Number of Spanning Trees in Cartesian Product of Graphs

• Autorzy: Jernej Azarija
• Języki publikacji: EN

Abstrakty

EN

Let G1 and G2 be simple graphs and let n1 = |V (G1)|, m1 = |E(G1)|, n2 = |V (G2)| and m2 = |E(G2)|. In this paper we derive sharp upper and lower bounds for the number of spanning trees τ in the Cartesian product G1 □G2 of G1 and G2. We show that: [...] and [...] . We also characterize the graphs for which equality holds. As a by-product we derive a formula for the number of spanning trees in Kn1 □Kn2 which turns out to be [...] .

Słowa kluczowe

EN

Cartesian product graphs; spanning trees; number of spanning trees; inequality

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2013
• Tom: 33
• Numer: 4
• Strony: 785-790

Daty

wydano

2013-09-01

online

2013-10-15

Twórcy

autor

Jernej Azarija

• Department of Mathematics, University of Ljubljana Jadranska 21, 1000 Ljubljana, Slovenia

Bibliografia

• [1] R.B. Bapat and S. Gupta, Resistance distance in wheels and fans, Indian J. Pure Appl. Math. 41 (2010) 1-13.[WoS]
• [2] Z. Bogdanowicz, Formulas for the number of spanning trees in a fan, Appl. Math. Sci. 16 (2008) 781-786.
• [3] F.T. Boesch, On unreliabillity polynomials and graph connectivity in reliable network synthesis, J. Graph Theory 10 (1986) 339-352. doi:10.1002/jgt.3190100311
• [4] R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Probab. 21 (1993) 1329-1371. doi:10.1214/aop/1176989121[Crossref]
• [5] G.A. Cayley, A theorem on trees, Quart. J. Math 23 (1889) 276-378. doi:10.1017/CBO9780511703799.010[Crossref]
• [6] M.H.S. Haghighi and K. Bibak, Recursive relations for the number of spanning trees, Appl. Math. Sci. 46 (2009) 2263-2269.
• [7] R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, 2nd Edition (CRC press, 2011).
• [8] G.G. Kirchhoff, ¨Uber die Aufl¨osung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strome gefhrt wird, Ann. Phy. Chem. 72 (1847) 497-508. doi:10.1002/andp.18471481202[Crossref]
• [9] B. Mohar, The laplacian spectrum of graphs, in: Graph Theory, Combinatorics, and Applications (Wiley, 1991).
• [10] R. Shrock and F.Y. Wu, Spanning trees on graphs and lattices in d dimensions, J. Phys. A 33 (2000) 3881-3902. doi:10.1088/0305-4470/33/21/303 [Crossref]

Identyfikatory

DOI

10.7151/dmgt.1698