A simple formula for the number of spanning trees of line graphs

Suppose G=(V,E) is a loopless graph and Sk(G) is the graph obtained from G by subdividing each of its edges k (k≥0) times. Let T(G) be the set of all spanning trees of G, L(Sk(G)) be the line graph of the graph Sk(G) and t(L(Sk(G))) be the number of spanning trees of L(Sk(G)). By using techniques fr...

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Veröffentlicht in:Journal of graph theory 2018-06, Vol.88 (2), p.294-301
Hauptverfasser: Gong, Helin, Jin, Xian'an
Format: Artikel
Sprache:eng
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Zusammenfassung:Suppose G=(V,E) is a loopless graph and Sk(G) is the graph obtained from G by subdividing each of its edges k (k≥0) times. Let T(G) be the set of all spanning trees of G, L(Sk(G)) be the line graph of the graph Sk(G) and t(L(Sk(G))) be the number of spanning trees of L(Sk(G)). By using techniques from electrical networks, we first obtain the following simple formula: t(L(Sk(G)))=1∏v∈Vd2(v)×∑T∈T(G)∏e=xy∈E(T)d(x)d(y)×∏e=uv∈E∖E(T)[d(u)+kd(u)d(v)+d(v)].Then we find it is in fact equivalent to a complicated formula obtained recently using combinatorial techniques in [F. M. Dong and W. G. Yan, Expression for the number of spanning trees of line graphs of arbitrary connected graphs, J. Graph Theory. 85 (2017) 74–93].
ISSN:0364-9024
1097-0118
DOI:10.1002/jgt.22212