Expression for the Number of Spanning Trees of Line Graphs of Arbitrary Connected Graphs

For any graph G, let t(G) be the number of spanning trees of G, L(G) be the line graph of G, and for any nonnegative integer r, Sr(G) be the graph obtained from G by replacing each edge e by a path of length r+1 connecting the two ends of e. In this article, we obtain an expression for t(L(Sr(G))) in terms of spanning trees of G by a combinatorial approach. This result generalizes some known results on the relation between t(L(Sr(G))) and t(G) and gives an explicit expression t(L(Sr(G)))=km+s-n-1(rk+2)m-n+1t(G) if G is of order n+s and size m+s in which s vertices are of degree 1 and the others are of degree k. Thus we prove a conjecture on t(L(S1(G))) for such a graph G. (C) 2016 Wiley Periodicals, Inc.