When (signless) Laplacian coefficients meet matchings of subdivision

Zhibin Du

Abstract: Let G be a graph, whose subdivision is denoted by S(G). Let ϕL(G,x) be the characteristic polynomial of the Laplacian matrix of G. In 1974, Kelmans and Chelnokov (1974) gave a graph theoretical interpretation for the coefficients of ϕL(G,x), in terms of the spanning forests of G. In this paper, we present another graph theoretical interpretation of the Laplacian coefficients by using the matching numbers of S(G), generalizing the cases of trees and unicyclic graphs, which were established by Zhou and Gutman (2008) and Chen and Yan (2021), respectively. Analogously, a graph theoretical interpretation of the signless Laplacian coefficients is also presented, whose previous graph theoretical interpretation is based on the so-called TU-subgraphs (the spanning subgraphs whose components are trees or odd-unicyclic graphs) due to Cvetković et al. (2007). Some formulas related to the number of spanning trees are also given.