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

Abstract

Suppose G=(V,E) is a loopless graph and Skfalse(Gfalse) 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(Skfalse(Gfalse)) be the line graph of the graph Skfalse(Gfalse) and t(Lfalse(Sk(G)false)) be the number of spanning trees of L(Skfalse(Gfalse)). By using techniques from electrical networks, we first obtain the following simple formula: truerightleftt(Lfalse(Sk(G)false))=1∏v∈Vd2false(vfalse)rightleft×∑T∈T(G)∏e=xy∈E(T)dfalse(xfalse)dfalse(yfalse)rightleft×∏e=uv∈E∖E(T)false[d(u)+kd(u)d(v)+d(v)false].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].

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