When (signless) Laplacian coefficients meet matchings of subdivision

Let G be a graph, whose subdivision is denoted by S ( G ). Let L ( 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 L ( 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 oddunicyclic graphs) due to Cvetkovi & cacute;et al. (2007). Some formulas related to the number of spanning trees are also given. (c) 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.