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Given a connected regular graph 
$G$, let 
$l(G)$ be its line graph, 
$s(G)$ its subdivision graph, 
$r(G)$ the graph obtained from 
$G$ by adding a new vertex corresponding to each edge of 
$G$ and joining each new vertex to the end vertices of the corresponding edge and 
$q(G)$ the graph obtained from 
$G$ by inserting a new vertex into every edge of 
$G$ and new edges joining the pairs of new vertices which lie on adjacent edges of 
$G$. A formula for the normalised Laplacian characteristic polynomial of 
$l(G)$ (respectively 
$s(G),r(G)$ and 
$q(G)$) in terms of the normalised Laplacian characteristic polynomial of 
$G$ and the number of vertices and edges of 
$G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of 
$l(G)$ (respectively 
$s(G),r(G)$ and 
$q(G)$).
