Combinatorial explanation of coefficients of the Laplacian matching polynomial of graphs

Mohammadian (J Alg Combin 52 (2020), 33-39) defined the Laplacian matching polynomial L-G(x) of a simple graph G and obtained many similar properties to the matching polynomial. In this paper, we extend the definition of L-G(x) to weighted graphs (vertex-weighted and edge-weighted graphs). Using the principle of inclusion and exclusion, we prove that the Laplacian matching polynomial of a weighted graph G is equivalent to the matching polynomial of a weighted subdivision S(G), which results in a combinatorial explanation of the coefficients of the Laplacian matching polynomial. (C)& nbsp;2022 Elsevier B.V. All rights reserved.