Enumerating perfect matchings in line graphs of graphs with pendent edges

Let G be a connected graph, and denote by α(G) the number of perfect matchings in G. Let L(G) (M(G)) be the line graph (middle graph) of G respectively. For a graph G′ constructed by adding a pendant edge to each vertex in G, Lai et al. [Discrete Mathematics, 347 (2024) 113847] proved that when the number of vertices in M(G) is even and the maximum degree of G is less than or equal to 4, then α(L(G′))=α(M(G))=2m−n+13m−n2, where m and n are the number of edges and vertices in G. The present work generalizes the above result by introducing a broader family of graphs {Gr→}. Suppose G is a graph with vertex set V(G)={u1,…,un} and let r→=(r1,…,rn) be a vector of positive integers. The graph Gr→ is obtained from G by adding ri pendant vertices adjacent to each vertex ui in G, and we denote r=∑i=1nri. We prove that: If |E(Gr→)|=m+r is even, then α(L(Gr→))≥2m−n+13m+r−2n2. When the maximum degree of Gr→ is less than or equal to 5, the equality holds. As applications, we provide explicit counting results for perfect matchings in chain and cyclic silicate structures.