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 G G contains at least 2|E|?|V|+1 2 | E | ? | V | + 1 perfect matchings, and characterize G G such that L(G) L ( G ) has exactly 2|E|?|V|+1 2 | E | ? | V | + 1 perfect matchings. As applications, we use a unified approach to solve the dimer problem on the Kagom lattice, 3.12.12 3.12.12 lattice, and Sierpinski gasket with dimension two in the context of statistical physics.
perfect matching line graph cyclomatic number