Hosoya polynomials under gated amalgamations

An induced subgraph HH of a graph GG is gated if for every vertex xx outside HH there exists a vertex x′x′ inside HH such that each vertex yy of HH is connected with xx by a shortest path passing through x′x′. The gated amalgam of graphs G1G1 and G2G2 is obtained from G1G1 and G2G2 by identifying their isomorphic gated subgraphs H1H1 and H2H2. Two theorems on Hosoya polynomials of gated amalgams are provided. As their applications, explicit expressions for Hosoya polynomials of hexagonal chains are obtained.