On the number of perfect matchings of line graphs

Let G=(V,E)G=(V,E) be a connected graph, where |E||E| is even. In this paper we show that the line graph L(G)L(G) of GG contains at least 2|E|−|V|+12|E|−|V|+1 perfect matchings, and characterize GG such that L(G)L(G) has exactly 2|E|−|V|+12|E|−|V|+1 perfect matchings. As applications, we use a unified approach to solve the dimer problem on the Kagomé lattice, 3.12.123.12.12 lattice, and Sierpinski gasket with dimension two in the context of statistical physics.