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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 ,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 article, 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(  r  k  +  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 .

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