Perfect Matchings of Line Graphs with Small Maximum Degree

Let G be a connected graph with vertex set V (G) = {v1, v2, · · · , vν} , which may have multiple edges but have no loops, and 2 ≤ dG(vi) ≤ 3 for i = 1, 2, · · · , ν, where dG(v) denotes the degree of vertex v of G. We show that if G has an even number of edges, then the number of perfect matchings of the line graph of G equals 2, where n is the number of 3-degree vertices of G. As a corollary, we prove that the number of perfect matchings of a connected cubic line graph with n vertices equals 2 if n > 4, which implies the conjecture by Lovász and Plummer holds for the connected cubic line graphs. As applications, we enumerate perfect matchings of the Kagomé lattices, 3.12.12 lattices, and Sierpinski gasket with dimension two in the context of statistical physics.