Expression for the Number of Spanning Trees of Line Graphs of Arbitrary Connected Graphs
Fengming Dong, Weigen Yan
备注：
22 pages, 7 pages, presented at the National Conference of Combinatorics and Graph Theory at Guangzhou, China in Nov. 2014. It was submitted to JGT in Feb. 2014 and revised in July 2015
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 non-negative integer $r$, $S_r(G)$ 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 paper we obtain an expression for $t(L(S_r(G)))$ in terms of spanning trees of $G$ by a combinatorial approach. This result generalizes some known results on the relation between $t(L(S_r(G)))$ and $t(G)$ and gives an explicit expression $t(L(S_r(G)))=k^{m+s-n-1}(rk+2)^{m-n+1}t(G)$ 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 $t(L(S_1(G)))$ for such a graph $G$.
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