On the number of spanning trees of some irregular line graphs

Let G be a graph with n vertices and m edges and Δ and δ the maximum degree and minimum degree of G. Suppose G ′ is the graph obtained from G by attaching Δ − d G ( v ) pendent edges to each vertex v of G. It is well known that if G is regular (i.e., Δ = δ , G = G ′ ), then the line graph of G, denoted by L ( G ) , has 2 m − n + 1 Δ m − n − 1 t ( G ) spanning trees, where t ( G ) is the number of spanning trees of G. In this paper, we prove that if G is irregular (i.e., Δ ≠ δ ), then t ( L ( G ′ ) ) = 2 m − n + 1 Δ m + s − n − 1 t ( G ) , where s is the number of vertices of degree one in G ′ .