Spanning Trees in Regular Graphs

Let X be a regular graph with degree k 2 3 and order n. Then the number of spanning trees of X is where yk, ck and (3,,k(l/k) are positive constants, and p, is the number of equivalence classes of certain closed walks of length i in X. The value (k-l)k-l Ck = (k2_2k)"i/2)-l is shown to be the best possible in the sense that K(x~)"" +c^ for some increasing sequence X,, X2,.. . of regular graphs of degree k. A sufficient condition for this convergence is established. Finally, for some absolute constant A, K (X) SAC: log n/(nk log k), a bound which (for fixed k) is high by at most O(log n). In this paper we investigate the number of spanning trees of a regular graph. We succeed in finding a tight upper bound in terms of the numbers of small cycles and other subgraphs. The only previous similar result known to the author was found by Kel'mans [7] and independently by Nosal [l31 and Biggs [2]: THEOREM 1.1. A regular graph of order n and degree k has at most (n k / (n-l)) " l / n spanning trees. We will not allow our graphs to have multiple edges, but the same results can easily be extended to that case also. A walk of length r in a graph X is a sequence v = (vo, 0 1 ,. .. , v,) of vertices of X such that is adjacent to vi for l G i S r. We say that v starts at v", finishes at vr, and is closed if v, = U". Suppose that for some i (0 < i <r) we have = vi+l. Then we can reduce v by deleting the elements vi and The result is clearly a walk of length r-2 which is closed if and only if v is closed. If v cannot be reduced in this way it is called irreducible. Given any walk v there is a unique irreducible walk i5 which can be obtained from v by a sequence of reductions. The uniqueness of 17 is proved in [ 5 ]. If I7 has length 0, we will call v totally reducible. Obviously, totally reducible walks are closed. Our first theorem gives a relationship between the number of walks and the number of irreducible walks between two vertices of X, if X is regular. THEOREM …