Enumeration of perfect matchings of the middle graph of a graph G with △(G)≤4

Abstract

Let G be a connected graph with n vertices and m edges, and L(G) and M(G) be the line graph and middle graph of G, respectively. Ciucu et al. (2012) [2] and Dong et al. (2013) [3] proved independently that if m is even and the maximum degree Δ(G) of G satisfies Δ(G)≤3, then L(G) has 2m−n+1 perfect matchings. Recently, Li et al. (2023) [6] showed that if G is a connected cubic graph and m is even, then the number of perfect matchings of M(G) equals 2n2+13n4. In this paper, we extend the results above and show that if G is a connected graph with △(G)≤4 and m+n is even, then the number of perfect matchings of M(G) equals 2m−n+13m−n2.